Geometry - NJCTLcontent.njctl.org/courses/math/geometry-2015-16/3d-geometry/3d... · 2 Geometry 3D...

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Geometry

3D Geometry

2015-10-28

www.njctl.org

http://www.njctl.org

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Table of Contents

Intro to 3-D SolidsViews & Drawings of 3-D SolidsSurface Area of a Prism

Surface Area of a CylinderSurface Area of a PyramidSurface Area of a Cone

Click on the topic to go to that section

Volume of a PrismVolume of a Cylinder

Volume of a PyramidVolume of a ConeSurface Area & Volume of SpheresCavaleri's PrincipleSimilar SolidsPARCC Sample Questions

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Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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Intro to 3-Dimensional Solids

Return to Table of Contents

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2-dimensional drawings use only the x and y axes

X

Y

Length

widthY

X

Length width

Y

X Length

width

Intro to 3-D Solids

7

Y

X

Z

height

height

Y

X

3-dimensional drawings include the x, y and z-axis.

The z-axis is the third dimension.

The third dimension is the height of the figure

Intro to 3-D Solids

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Y

X

Z

height

height

YX

x

Y

Intro to 3-D Solids

9

Y

X

Z

height

Y

X

X

Y

r

Intro to 3-D Solids

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To give a figure more of a 3-dimensional look, lines that are not visible from the angle the figure is being viewed are drawn as dashed line segments. These are called hidden lines.

Y

X

Z

height

height

Intro to 3-D Solids

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A Polyhedron (pl. Polyhedra) is a solid that is bounded by polygons, called faces. An edge is the line segment formed by the intersection of 2 faces. A vertex is a point where 3 or more edges meet

Face

Edge Vertex

Intro to 3-D Solids

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The 3-Dimensional Figures discussed in this unit are:

Pyramids

CylindersPrisms

Intro to 3-D Solids

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The 3-Dimensional Figures discussed in this unit are:

. C

Cones: Spheres:

Intro to 3-D Solids

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Right Vs. Oblique

Right Right

In Right Prisms & Cylinders, the bases are aligned directly above one another. The edges are perpendicular with both bases.

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Right Vs. ObliqueIn Oblique Prisms & Cylinders, the bases are not aligned directly above one another. The edges are not perpendicular with the bases.

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Right Vs. Oblique

Right Oblique

Right Oblique

In Right Pyramids & Cones, the vertex is aligned directly above the center of the base.

In Oblique Pyramids & Cones, the vertex is not aligned

directly above the center of the base.

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Prisms have 2 congruent polygonal bases. The sides of a base are called base edges.

The segments connecting corresponding vertices are lateral edges.A

B

C

X Y

ZIn this diagram:There are 6 vertices: A, B, C, X, Y, & ZThere are 2 bases: ABC & XYZ.There are 6 base edges: AB, BC, AC, XY, YZ, & XZ.There are 3 lateral edges: AX, BY, & CZ.This prism has a total of 9 edges.

Intro to 3-D Solids

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The polygons that make up the surface of the figure are called faces. The bases are a type of face and are parallel and congruent to each other. The lateral edges are the sides of the lateral faces.

A B

C

X Y

Z

In this diagram:There are 2 bases: ABC & XYZ.

There are 3 lateral faces: AXBY, BYCZ, & CZAX.

This prism has a total of 5 faces.

Intro to 3-D Solids

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1

A AB

B DE

C FS

D CP

E FA

F CD

G NP

H BC

I DQ

AB C

DEF

M

N PQ

RS

Choose all of the base edges.

Ans

wer

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2

A AB

B CD

C ER

D BN

E DQ

F QR

G MS

H AM

I CP

Choose all of the lateral edges.

AB C

DEF

M

N PQ

RS Ans

wer

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3 Chooses all of the bases.

A AFSM

B FERS

C EDQR

D ABCDEF

E CDQP

F BCPN

G MNPQRS

H ABNM

Ans

wer

AB C

DEF

M

N PQ

RS

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4 Chooses all of the lateral faces.

A AFSM

B FERS

C EDQR

D ABCDEF

E CDQP

F BCPN

G MNPQRS

H ABNM

AB C

DEF

M

N PQ

RS Ans

wer

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5 Chooses all of the faces.

A AFSM

B FERS

C EDQR

D ABCDEF

E CDQP

F BCPN

G MNPQRS

H ABNM

AB C

DEF

M

N PQ

RS Ans

wer

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A pyramid has 1 base with vertices and the lateral edges go to a single vertex.

A

M

N P

RS

Q

This pyramid has:6 lateral edges,6 base edges,12 edges (total)

7 vertices

Intro to 3-D Solids

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A pyramid has faces that are polygons: 1 base and triangles that are the lateral faces.

A

M

N PQ

RS

This pyramid has:6 lateral faces,1 base,7 faces (total)

Intro to 3-D Solids

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6 Choose all of the base edges.

A VN

B KN

C VL

D LM

E VM

F VK

K

L

MN

V

G KL

H NM

Ans

wer

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7 Choose all of the lateral edges.

A VN

B KN

C VL

D LM

E VM

F VK

G KL

H NM

K

L

MN

V

Ans

wer

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8 How many edges does the pyramid have?

Ans

werK

L

MN

V

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9 Choose all of the lateral faces.

A KNV

B NMV

C KLMN

D VML

E KLV

K

L

MN

V

Ans

wer

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10 Choose all of the bases.

A KNV

B NMV

C KLMN

D VML

E KLV

K

L

MN

V

Ans

wer

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11 How many faces does the pyramid have?

Ans

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K

L

MN

V

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.

.

A

B

A cylinder has 2 bases which are congruent circles. The lateral face is a rectangle wrapped around the circles.

A & Bare the bases

of the cylinder.

Intro to 3-D Solids

A cylinder can also be formed by rotating a rectangle about an axis.

Click for sample animation

https://tube.geogebra.org/student/m18768

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A cone, like a pyramid, has one base which is a circle.

. N

V

N is thebase of the cone.

V is the vertex of the cone.

Intro to 3-D Solids

A cone can also be formed by rotating a right triangle about one of its legs.

Click for sample animation

http://www.mathsisfun.com/geometry/cone.html

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A sphere is a 3-dimensional circle in that every point on the sphere is the same distance from the center.

. C

Similar to a circle, a sphere is named by its center point. Sphere C is the solid shown above.

Intro to 3-D Solids

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12 Which solids have 2 bases?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

Ans

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13 Which solid has one vertex?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

Ans

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14 Which solid has more base edges than lateral edges?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

Ans

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15 Which solid(s) have no vertices?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

Ans

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16 Which solid is formed when rotating an isosceles triangle about its altitude?

A a prism

B a cylinder

C a pyramid

D a cone

E a sphere

Ans

wer

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Euler's Theorem states that the number of faces (F), vertices (V), and edges (E) satisfy the formula F + V = E + 2

AB

C

X Y

Z

A

MN P

QRS

F = 5V = 6E = 9

5 + 6 = 9 + 211 = 11

F = 7V = 7

E = 127 + 7 = 12 + 2

14 = 14

Intro to 3-D Solids

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Example:A solid has 12 faces, 2 decagons and 10 trapezoids. How many vertices does the solid have?

V + F = E + 2V + 12 = 30 + 2V + 12 = 32V = 20

On their own, the 2 decagons & 10 trapezoids have2(10) + 10(4) = 60 edges. In a 3-D solid, each side is shared by 2 polygons. Therefore, the number of edges in the solid is 60/2 = 30.

Intro to 3-D Solids

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click

click

click

click

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Example:A solid has 9 faces, 1 octagon and 8 triangles. How many vertices does the solid have?

V + F = E + 2V + 9 = 16 + 2V + 9 = 18V = 9

What information do you have?9 faces & the 2 types of faces

Intro to 3-D Solids

click

click

click

click

What is the problem asking? Create an equation to represent the problem.

How are the number of edges in the 2-D faces, related to the number of edges in the polyhedron? Write a number sentence to describe this situation.

(1(8) + 8(3))/2(8 + 24)/2

32/216 edges

click

click

click

click

click

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17 A solid has 10 faces, one of them being a nonagon and 9 triangles. How many vertices does it have?

A 8

B 9

C 10

D 18

Ans

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C

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18 A solid has 12 faces, all of them being pentagons. How many vertices does it have?

A 30

B 20

C 15

D 10

Ans

wer B

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19 A solid has 8 faces, all of them being triangles. How many vertices does it have?

A 24

B 12

C 8

D 6 Ans

wer

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A cross-section is the locus of points of the intersection of a plane and a 3-D solid.

Cross-SectionIntro to 3-D Solids

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Think about it as if the plane were a knife and you were cutting the shape, what would the cut look like?

Cross-Section

Circle Ellipse

Parabola (with the inner section shaded)

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Cross-sections of a surface are a 2-dimensional figure.

Cross-sections of a solid are a 2-dimensional figure and its interior.

The top can be removed to see the cross section. (Try it out)

Cross-Section

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20 What is the locus of points (cross-section) of a cube and a plane perpendicular to the base and parallel to the non-intersecting sides?

A square

B rectangle

C trapezoid

D hexagon

E rhombus

F parallelogram

G triangle

H circle

Ans

wer

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21 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane perpendicular to the base and parallel to the non-intersecting sides?

A 72 sq inches

B 144 sq inches

C 187.06 sq inches

D 203.65 sq inches 12 in. Ans

wer

B

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22 What is the locus of points of a cube and a plane that contains the diagonal of the base and is perpendicular to the base?

A square

B rectangle

C trapezoid

D hexagon

E rhombus

F parallelogram

G triangle

H circle

Ans

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B

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23 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane that contains the diagonal of the base and is perpendicular to the base?

A 72 sq inches

B 144 sq inches

C 187.06 sq inches

D 203.65 sq inches

12 in. Ans

wer

D

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24 What is the locus of points of a cube and a plane that contains the diagonal of the base but does not intersect the opposite base?

A square

B rectangle

C trapezoid

D hexagon

E rhombus

F parallelogram

G triangle

H circle

Ans

wer

G

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25 What is the locus of points of a cube and a plane that intersects all of the faces?

A square

B rectangle

C trapezoid

D hexagon

E rhombus

F parallelogram

G triangle

H circle

Ans

wer

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Views & Drawings of 3-D Solids


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Isometric drawings are drawings that look 3-D & are created on a grid of dots using 3 axes that intersect to form 120° & 60° angles.

Views & Drawings

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Example: Create an Isometric drawing of a cube.

Ans

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Views & Drawings

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An Orthographic projection is a 2-D drawing that shows the different viewpoints of an object, usually from the front, top & side. Each drawing depends on your position relative to the figure.

Front Side

Top (from front)

Views & Drawings

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Consider these three people viewing a pyramid:

Views & Drawings

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The orange person is standing in front of a face, so their view is a triangle.

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The green person is standing in front of a lateral edge, so from their view they can see 2 faces.

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The purple person is flying over and can see the four lateral faces.

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26 Given the surface shown, what would be the view from point A?

A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (front) Ans

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (top)

Ans

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (top)

right square prism

Ans

wer

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a trapezoid

A (front)

right square prism

Ans

wer

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (front)

Ans

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (above)

Ans

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (above)

Ans

wer

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A (front)

Ans

wer

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A a Rectangle

B a Square

C a Circle

D a Pentagon

E a Triangle

F a Parallelogram

G a Hexagon

H a Trapezoid

A

sphere

Ans

wer

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AB

C (Looking down from above)What would the view be like from each position?

Views & Drawings

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A

What would the view be like from each position?

From A, how many columns of blocks are visible?

- 3 columns How tall is each column?

- first one is 4 high- second & third columns are each 2 blocks high

Click to reveal

Click to reveal

Views & Drawings

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B


From B, how many columns of blocks are visible?

- 2 columns

How tall is each column?- left one is 3 high- right one is 4 high

Click to reveal

Click to reveal

Views & Drawings

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C (Looking down from above)


From C, how many columns of blocks are visible?

- 3 columns How tall is each column?

- all of them are 2 blocks high

Click to reveal

Click to reveal

Views & Drawings

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FrontSide

AboveDraw the 3 views.

Side ViewTop View

Front View

Move for Answer

Views & Drawings

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FrontSide

AboveDraw the 3 views.

AboveFront Side

Views & Drawings

Move for Answer

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Here are 3 views of a solid, draw a 3-dimensional representation.

Top FrontSide

L R

F

Views & Drawings

Move for Answer Mat

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Here are 3 views of a solid, draw a 3-dimensional representation.

TopF

L R

Side Front

Views & Drawings

Move for Answer Mat

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Surface Area of a Prism


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A Net is a 2-dimensional shape that folds into a 3-dimensional figure.

The Net shows all of the faces of the surface.

Net

6

646 4

12

4

Shown is the net of a right rectangular prism.

12

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Teac

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The PARCC Reference sheet for the HS level does NOT contain any formulas to calculate the Surface Area (reference sheet is linked to this pull tab - just click on it to download it). Encourage students to either memorize future formulas, or draw the net each time so that they can break down the solid into smaller 2-D shapes.

http://www.parcconline.org/files/119/Mathematics%20Guidance%20Policies/245/Operational%20PARCC%20High%20School%20Math%20Reference%20Sheet.pdf

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The net shown is a right triangular prism. The lateral faces are rectangles. The bases are on opposite sides of the rectangles, although they do not need to be on the same rectangle.

Net

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The nets shown are for the same right triangular prism. Net

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Nets of oblique prisms have parallelograms as lateral faces.

Nets

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Rectangular Prisms

cube

ww w

H

HH

ℓ ℓ ℓ

Teac

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To avoid confusion with the "heights" when calculating the surface area of a triangular prism, the height of the prism has been assigned "H". The triangular height will be "h" starting on slide #98

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Base

Base height

Base

height

Base

A prism has 2 bases.

The base of a rectangular prism is a rectangle.

The height of the prism is the length between the two bases.

Rectangular Prisms

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The Surface Area of a figure is the total amount of area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift).

Area

Area

Area

Area

AreaArea

Top Area

Side

Area Front Area Bottom Area

Back Area

Side

Area

The Surface Area of a figure is the sum of the areas of each side of the figure.

Rectangular Prisms

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Finding the Surface Area of a Rectangular Prism

H

wℓ

Area of the Top = ℓ x w

Area of the Bottom = ℓ x w

Area of the Front = ℓ x H

Area of the Back = ℓ x H

Area of Left Side = w x H

Area of Right Side = w x H

The Surface Area is the sum of all the areas

S.A. = ℓw + ℓw + ℓH + ℓH + wH + wH

S.A. = 2 ℓw + 2 ℓH + 2wH

Teac

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Example: Find the surface area of the prism

74

3

Area of Top & Bottom Area of Right & Left

Area of Front & Back

A = 7(4) = 28u2 A = 3(4) = 12 u2

A = 3(7) = 21 u2

Click

Click

Click

Total Surface Area = 2(28) + 2(12) + 2(21) = 56 + 24 + 42 = 122 units2

Click

Click

Finding the Surface Area of a Rectangular Prism

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35 What is the total surface area, in square units?

4

5

9

Ans

wer SA = 2(4)(5) + 2(4)(9) + 2(5)(9)

SA = 40 + 72 + 90SA = 202 sq. units

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36 What is the total surface area, in square units?

8

8

8

Ans

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37 Troy wants to build a cube out of straws. The cube is to have a total surface area of 96 in2, what is the total length of the straws, in inches?

Ans

wer

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S.A. = 2B + PH

The Surface Area is the sum of the areas of the 2 Bases plus the Lateral Area (Perimeter of the base, P, times the height of the prism, H)

The Lateral Area is the area of the Lateral Surface. The Lateral Surface is the part that wraps around the middle of the figure (in between the two bases).

Another Way of Looking at Surface Area

Lateral SurfaceBas

eBas

e

Base

Base

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Base

Base

w

H

ℓ

Another formula for Surface Area of a right prism: S.A. = 2B + PH

B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism

S.A. = 2B + PH

S.A. = 2 ℓw + (2 ℓ +2w)H

S.A. = 2 ℓw + 2 ℓH + 2wH

Rectangular Prisms

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Base

Base

w

H

ℓ

In the surface area formula, 2B is the sum of the area of the 2 bases.

What does PH represent? The area of lateral faces or Lateral AreaClick

Rectangular PrismsAnother formula for Surface Area of a right prism:

S.A. = 2B + PH

B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism

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38 If the base of the prism is 12 by 6, what is the lateral area, in sq ft?

12 ft6 ft

4 ft

Ans

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39 The surface area of the rectangular prism is :

A 24 sq ft

B 144 sq ft

C 288 sq ft

D 48 sq ft

E 72 sq ft

12 ft6 ft

4 ft

Ans

wer

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40 If 7 by 6 is base of the prism, what is the lateral area, in sq units?

7

96 A

nsw

er

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41 What is the total square units of the surface area?

7

96

Ans

wer

100

42 Find the value of y, if the lateral area is 144 sq units, and y by 6 is the base.

y

6 8

Ans

wer

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43 What is the value of the missing variable if the surface area is 350 sq. ft.

A 7 ft

B 8.3 ft

C 12 ft

D 15 ft

x ft5 ft

10 ft

Ans

wer

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44 Sharon was invited to Maria's birthday party. For a present, she purchased an iHome (a clock radio for an iPod or iPhone) which is contained in a box that measures 7 inches in length, 5 inches in width, and 4 inches in height. How much wrapping paper does Sharon need to wrap Maria's present?

Ans

wer SA = 2(5)(7) + 2(5)(4) + 2(7)(4)

= 70 + 40 + 56SA = 166 sq. inches

103

Other Prisms

104

base

base height

base

base height

base

base

height

basebase height

A Prism has 2 Bases

The Base of a Prism matches the first word in the name of the prism. e.g. the Base of a Triangular Prism is a Triangle

The Height of the Prism is the length between the two bases

Other Prisms

105

The Surface Area of a figure is the total amount of Area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift).

The Surface Area of a figure is the sum of the areas of each side of the figure

Area AreaArea

Area

Area

Area AreaArea

Area Area

Other Prisms

Triangular PrismNet of the Triangular Prism

106

Finding the Surface Area of a Right PrismSurface Area: S.A. = 2B + PH B = Area of the triangular base = ½bh P = Perimeter of the triangular base = a + b + c H = Height of the prism

Lateral Area = PH = (a + b + c)H

The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases.

base

basePrism's

height

a

b

cH

P = a + b + c

ac

bc a

Lateral SurfaceHh

bB = ½ bh

Note: The formula above will work for any right prism.

107

Example: Find the lateral area and surface area of the right triangular prism.

10

611

Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem.62 + b2 = 102

36 + b2 = 100b2 = 64 b = 8 units

Next, calculate the perimeter of your base.P = 6 + 8 + 10 = 24 units

Use this to find the Lateral AreaLA = PH = 24(11) = 264 units2

Other Prisms

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you do this mentally? (MP5)- Referring to the Pythagorean TripleCan you find a shortcut to solve the problem? How would the shortcut make the problem easier? (MP8)- Referring to the Pythagorean TripleWhat labels could you use? (MP6)

108

10

611

Example: Find the lateral area and surface area of the right triangular prism.

Then, calculate the area of your base, BB = (1/2)(8)(6) = 24 units2

Finally, calculate your Surface Area.SA = 2B + PHSA = 2(24) + (24)(11)SA = 48 + 264 = 312 units2

Other Prisms

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Example: Find the lateral area and surface area of the triangular prism. 99

9

12

Since it has a base that is an equilateral triangle, we need to find the height of the triangle using Pythagorean Theorem or the 30-60-90 Triangle Theorem.4.52 + b2 = 92

20.25 + b2 = 81b2 = 60.75 b = 4.5√3 units = 7.79 units

Next, calculate the perimeter of your base.P = 9 + 9 + 9 = 27 units

Use this to find the Lateral AreaLA = PH = 27(12) = 324 units2

Other Prisms

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you do this mentally? (MP5)- Referring to the 30-60-90 triangleCan you find a shortcut to solve the problem? How would the shortcut make the problem easier? (MP8)- Referring to the 30-60-90 triangleWhat labels could you use? (MP6)

110

Example: Find the lateral area and surface area of the triangular prism.

Then, calculate the area of your base, BB = (1/2)(9)(4.5√3) = 20.25√3 units2 = 35.07 units2

Finally, calculate your Surface Area.SA = 2B + PHSA = 2(35.07) + (27)(12)SA = 70.14 + 324 = 394.14 units2

Other Prisms

99

9

12

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45 The height of the triangular prism below is 11 ft, the height of the base is 3 ft, and the triangular base is an isosceles triangle. Find the surface area.

A 88 sq ft

B 132 sq ft

C 198 sq ft

D 222 sq ft 3 ft5 ft

11 ft

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D(1/2) of the base in the triangle is 4...3-4-5 Pyth. Triple, so the base of the triangle is 8.

P = 5 + 5 + 8 = 18B = (1/2)(8)(3) = 12SA = 2(12) + 18(11)SA = 24 + 198SA = 222 sq ft

112

46 The height of the triangular prism below is 3, and the triangular base is an equilateral triangle. Find the surface area.

A 64 sq ft

B 127.43 sq ft

C 72 sq ft

D 55.43 sq ft 8 ft3 ft

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BThe height of the triangular base is 4√3 = 6.93...30-60-90 triangle.

P = 8(3) = 24B = (1/2)(8)(4√3) = 16√3SA = 2(16√3) + 24(3)SA = 32√3 + 72SA = 127.43 sq ft

113

47 Find the lateral area of the right prism.

5

56

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hypotenuse of the right triangle is 5√2...45-45-90 triangle

P = 5 + 5 + 5√2 = 10 + 5√2 = = 17.07LA = (10 + 5√2)6LA = 60 + 30√2LA = 102.43 sq units

114

Finding the Surface Area of a Right PrismSurface Area : S.A. = 2B + PH B = Area of the regular hexagonal base = ½aP - a is the apothem of the regular base P = Perimeter of the base = b + c + d + e + f + g H = Height of the prism = HLateral Area = PH = (b + c + d + e + f + g)H

a

B = ½ aP

g

cb

H

e

f cde

fb

d

P = b + c + d + e + f + g

base

base

Prism's height

115

a

B = ½ aP

Finding the Surface Area of a Right Prism

P = b + c + d + e + f + g

base

base

Prism's height

g

cb

H

e

f cde

fb

d

The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases.

116

8 in

7 in

30°

4 in.

a

Example: Find the lateral area and surface area of the regular hexagonal prism.

Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle.

= 60° = central angle

= 30° = top angle of the triangle.

360 6

60 2

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When calculating the surface area and/or volume of regular polygonal prisms/pyramids, students could also find the measurement of the 2 congruent base angles using the Triangle Sum Theorem.

Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)Can you do this mentally? (MP5)What labels could you use? (MP6)

117


Next find the apothem using trigonometry, or special right triangles (if it applies).

tan 30 =

atan30 = 4 tan30 tan30

4 a

a = 4√3 = 6.93 in.

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118

B = (1/2)aP = (1/2)(4√3)(48) = 96√3 in2 = 166.28in2


Next, calculate the perimeter of your base.P = 8(6) = 48 inUse this to find the Lateral AreaLA = PH = 48(7) = 336 in2

Then, calculate the area of your base, B

Finally, calculate your Surface Area.SA = 2B + PHSA = 2(166.28) + (48)(7)SA = 332.56 + 336 = 668.56 in2

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8 in

7 in

119

36°

3 in.

a

Example: Find the lateral area and surface area of the right prism.

The base is a regular pentagon.

6 ft

10 ft

Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle.



360 5

72 2

Other Prisms

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120



Next find the apothem using trigonometry, or special right triangles (if it applies).

tan 36 = 3 a

atan36 = 3 tan36 tan36

a = 4.13 in.

6 ft

10 ft

Other Prisms

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121



Next, calculate the perimeter of your base.P = 5(6) = 30 in

Use this to find the Lateral AreaLA = PH = 30(10) = 300 in2

Then, calculate the area of your base, BB = (1/2)aP = (1/2)(4.13)(30) = 61.95 in2

Finally, calculate your Surface Area.SA = 2B + PHSA = 2(61.95) + (30)(10)SA = 123.9 + 300 = 423.9 in2

6 ft

10 ft

Other Prisms

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122


8 3

7

6

5Angles are right angles.

First, calculate the perimeter of your base. P = 8 + 7 + 5 + 4 + 3 + 3 P = 30 unitsUse this to find the Lateral AreaLA = PH = 30(6) = 180 units2

Other Prisms

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Then, calculate the area of your base, BB = 7(5)+3(3) = 44 units2

Finally, calculate your Surface Area.SA = 2B + PHSA = 2(44) + (30)(6)SA = 88 + 180 = 268 units2

123

48 Find the lateral area of the right prism.

8

11

The base is a regular hexagon.

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124

49 Find the total surface area of the right prism.


8

11

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need to start by finding the apothem = 4√3...30-60-90 triangleP = 6(8) = 48B = (1/2)(4√3)(48) = 96√3SA = 2(96√3) + 48(11)SA = 192√3 + 528 sq unitsSA = 860.55 sq units

125

50 Find the total surface area of the right prism.

4

4 3

2

10

9

All angles are right angles.

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P = 4 + 2 + 2 + 2 + 4 + 3 + 10 + 3P = 30B = 4(3) + 4(3) + 2(1) = 26SA = 2(26) + 30(9)SA = 52 + 270SA = 322 sq units

126

y

56

51 The right triangular prism has a surface area of 150 sq ft. Find the height of the prism.

A 5 ftB 6 ftC 7.81 ftD 6.38 ft

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127

Surface Area of a Cylinder


128

height

radius

base

base

height

radi

us

base base

Cylinders

A Cylinder is a solid w/ 2 circular bases that lie in || planes. Because each base is a circle, it contains a radius. The remaining measurement that connects the 2 bases is the height of the cylinder.

129

8

radius

The net of a right cylinder is two circles and a rectangle that forms the lateral surface.

8

x

What is the length of x?- The circumference of the circle (base)

radius

Click to reveal

Cylinders

130

Base

Base

height

Base

height Lateral Surface

Base

Finding the Surface Area of a Right Cylinder

Surface Area : S.A. = 2B + PH B = Area of the circular base = πr2 C = Perimeter of the Circular base (Circumference) = 2πr H = Height of the prism

Lateral Area = CH = 2πrH

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The PARCC Reference sheet for the HS level does NOT contain any formulas to calculate the Surface Area (reference sheet is linked to this pull tab - click on tab). Encourage students to either memorize the formulas, or draw the net each time so that they can break down the solid into smaller 2-D shapes.


131

Base

Base

height

Base

height Lateral Surface

Base


The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the cylinder between the circular bases.

Therefore, the Surface Area of a Cylinder can be simplified to the equation below. SA = 2πr2 + 2πrH

132

8

r = 4

Example: Find the lateral area and surface area of the right cylinder.

LA = 2πrhLA = 2π(4)(8)LA = 64π units2

LA = 201.06 units2

SA = 2πr2 + 2πrhSA = 2π(4)2 + 2π(4)(8)SA = 32π + 64πSA = 96π units2

SA = 301.59 units2

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133

34

d = 16

SA = 2πr2 + 2πrhSA = 2π(8)2 + 2π(8)(30)SA = 128π + 480πSA = 608π units2

SA = 1,910.09 units2

Example: Find the lateral area and surface area of the right cylinder.

LA = 2πrhLA = 2π(8)(30)LA = 480π units2

LA = 1507.96 units2

162 + h2 = 342

256 + h2 = 1156h2 = 900h = 30Note: 16-30-34 = 2(8-15-17) Pyth. Tripleclick

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134

Example: Find the lateral area and surface area of the right cylinder when the base circumference is 16π ft & the height is 10 ft.

SA = 2πr2 + 2πrhSA = 2π(8)2 + 2π(8)(10)SA = 128π + 160πSA = 288π ft2

SA = 904.78 ft2

LA = 2πrhLA = 2π(8)(10)LA = 160π ft2

LA = 502.64 ft2

C = 2πr16π = 2πr 2π 2π8 ft = r

Cylinders

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)Would it help to draw a picture? (MP4 & MP5) What tools do you need? (MP5)Can you do this mentally? (MP5)What labels could you use? (MP6)

135

h = 12

r = 7

52 Find the lateral area of the right cylinder.

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136

h = 12

r = 7

53 Find the surface area of the right cylinder. Use 3.14 as your value of π & round to two decimal places.

A 1200 sq in.B 307.72 sq in.C 835.24 sq in.D 1670.48 sq in. A

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137


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13

r = 5

138

h = 12


Base area is 36π units2

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139

h = 12

56 Find the surface area of the right cylinder.

Base area is 36π units2

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140

r = 8 in.

h

57 The surface area of the right cylinder is 653.12 sq in. Find the height of the cylinder. Use 3.14 as your value of π.

A 7 in.B 8 in.C 5 in.D 6 in.

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141

58 A food company packages soup in aluminum cans that have a diameter of 2 1/2 inches and a height of 4 inches. Before shipping the cans off to the stores, they add their company label to the can which does not cover the top and bottom. If the company is shipping 200 cans of soup to one store, how much paper material is required to make the labels?

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142

59 Maria's mom baked a cake for her daughter's birthday party. The diameter of the cake is 9 inches and the height is 2 inches.

How much base frosting (pink in the picture below) was required to cover the cake?

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143

Surface Area of a Pyramid


144

A Pyramid is a polyhedron in which the base is a polygon & the lateral faces are triangles with a common vertex.

Lateral Edges are the intersection of 2 lateral faces Vertex

LateralFace

LateralEdge

Base

Pyramids

145

Net

This is a right square pyramid. Another name for it is pentahedron.Hedron is a suffix that means face. Why is this a pentahedron?

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146

Slant Height

ℓ

The Pyramid has a square base and 4 triangular facesThe triangular faces are all isosceles triangles if its a right pyramid.The Height of each triangular face is the Slant Height of the pyramid if it is a regular pyramid (labeled as , or a cursive lower case L).

Surface Area = Sum of the Areas of all the sides

ℓ

Heightof theTriangle

147

Square Base (B)

Slant Height ( )ℓ

Pyramid's Height (h)

Segment Lengths in a Pyramid

148

Example: Find the value of x.

a2 + 122 = 132

a2 + 144 = 169a2 = 25a = 5Note: 5-12-13 Right Triangle

Therefore x = 2(5) = 10x

1312

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you do this mentally? (MP5)- Referring to the Pythagorean Triple: 5-12-13Can you find a shortcut to solve this problem? How would your shortcut make the problem easier? (MP8)- Referring to the Pythagorean Triple


149

Example: Find the value of x.

Base Area of the right square pyramid is 64 u2.

x8

Square Base has an area of 64, so64 = y2

y = 8, so a = 4 of the right triangle.

42 + 82 = x2

16 + 64 = x2

x2 = 80x = 8.94 units

Mat

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150

Example: Find the length of the slant height.

r

This is a regular hexagonal pyramid.

r = 6lateral edge = 12


151

First, find the height of the pyramid using Pythagorean Theorem.

h12

6

62 + h2 = 12236 + h2 = 144h2 = 108h = 6√3 = 10.39

Note: 30-60-90 triangler


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Mat

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you do this mentally? (MP5)- Referring to the 30-60-90 triangleCan you find a shortcut to solve this problem? How would your shortcut make the problem easier? (MP8)- Referring to the 30-60-90 triangle

152

Second, find the apothem of the hexagonal base.

a6 6

3 3

32 + a2 = 62

9 + a2 = 36a2 = 27a = 3√3 = 5.20Note: 30-60-90 triangle

= 60° = central

Note: equilateral

= 30° = top of the .

360 6

60 2r

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153

(3√3)2 + (6√3)2 = 2

27 + 108 = 2

2 = 135 = 3√15 = 11.62

ℓℓ

ℓℓ

Last, find the slant height of your pyramid w/ the apothem & height.

a = 3√3

ℓ h = 6√3

rclick

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Mat

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e Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)

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154

60 Find the value of the variable.

16

x6

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155


12

11x

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156


x6 Ans

wer

area of the base is 36 u2

157

63 Find the value of the slant height.

r

r = 8

lateral edge = 17

Regular Hexagonal Pyramid

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158

64 Find the value of the slant height.

a

a = 9

lateral edge = 12

Regular Hexagonal Pyramid

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10.82apothem in hexagon = 99 = x√3, so x = 3√3 & r = 6√3(6√3)2 + h2 = 122

h = 662 + 92 = 2 = 10.82

159

Square Base (B)

Slant Height ( )


ℓ

Surface Area = B + ½P and Lateral Area = ½P = Slant HeightP = Perimeter of BaseB = Area of Base

Surface Area of a Regular Pyramid

ℓ ℓℓ

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160

Square Base (B)

Slant Height ( )


Why is the Surface Area SA = B + P ? 1 2

Surface Area is the sum of all of the areas that make up the solid. In our diagram, these are 4 triangles & 1 square.Asquare = s s = s2 = B

A∆ = s 1 2 ℓ

ℓ

ℓ


161

Why is the Surface Area SA = B + P ? 1 2

Since there are 4 ∆s, we can multiply the area of each ∆ by 4. Therefore, our Surface Area for the Pyramid above isSA = s2 + 4(1/2)sSA = s2 + (1/2)(4s)SA = B + 1/2 P

s

ℓ

Net of Pyramid

ℓℓ

ℓ


ℓ

162

ℓ = 7

s = 6

Example: Find the lateral area and the surface area of the pyramid.

LA = 1/2 P ℓLA = 1/2 (24)(7)LA = 12(7)LA = 84 units2

SA = B + 1/2 P ℓSA = 62 + 1/2 (24)(7)SA = 36 + 84SA = 120 units2

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163


First, calculate the slant height.32 + 82 = ℓ 29 + 64 = ℓ273 = ℓ2 ℓ = 8.54Next, calculate the LA & SA

LA = 1/2 P ℓLA = 1/2 (24)(8.54)LA = 12(8.54)LA = 102.48 units2

SA = B + 1/2 P ℓSA = 62 + 1/2 (24)(8.54)SA = 36 + 102.48SA = 138.48 units2

h = 8

s = 6

Mat

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)What labels could you use? (MP6)


164


10

8

ℓ

First, calculate the slant height.

82 + ℓ 2 = 102

64 + ℓ 2 = 100 ℓ 2 = 36 ℓ = 6

s = 16

e = 10

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165


LA = 1/2 P ℓLA = 1/2 (64)(6)LA = 32(6)LA = 192 units2

SA = B + 1/2 P ℓ SA = 162 + 1/2 (64)(6)SA = 256 + 192SA = 448 units2

Next, calculate the LA & SA

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s = 16

e = 10


166


a

a = 4lateral edge = 8

Regular Pentagonal Pyramid


167

72°

36°36°4

x

r

Example: Find the lateral area and the surface area of the pyramid.First, find the radius & side length of the regular pentagon using the apothem & trigonometric ratios

tan36 =

x = 4tan36 = 2.91

Therefore, s = 2(2.91) = 5.82

= 36° = top of the .

360 5

72 2

x 4

4 rcos36 =

rcos36 = 4 cos36 cos36

r = 4.94

= 72° = centralClick

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168

Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pythagorean Theorem.

8

2.91

2.912 + ℓ 2 = 82

8.4681 + ℓ 2 = 64 ℓ 2 = 55.5319 ℓ = 7.45

ℓclick

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169

Last, find the lateral area & surface area of the pyramid.

LA = 1/2 P ℓLA = 1/2 (29.1)(7.45)LA = 108.40 units2

SA = B + 1/2 P ℓSA = 1/2 (4)(29.1) + 1/2 (29.1)(7.45)SA = 58.2 + 108.40SA = 166.6 units2 click

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170

65 Find the lateral area of the right pyramid.

s = 10

ℓ = 9

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171

66 Find the surface area of the right pyramid.

s = 10

ℓ = 9

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172


base

e = 10

area = 16

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173


base

e = 10

area = 16

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174

a

a = 5h = 12

Regular Octagonal Pyramid


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175

a

a = 5h = 12



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SA = B + 1/2 P ℓSA = 82.8 + 215.28SA = 298.08 units2

ℓ = 13 unitsP = 33.12 units

B = 1/2aP ℓB = 1/2(5)(33.12)B = 82.8 units2

176

Hint: The pyramid is NOT regular. So, B + 1/2 P ℓ doesn't work. Instead, draw a net of the

pyramid & find each area.


30

12

8

Hint

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177

Surface Area of a Cone


178

r

height

Slant Height ℓ

Lateral SurfaceSlant Height ℓ

Base

The Base of the cone is a circle

The length of the circular portion of the Lateral Surface is the same as the Circumference of the Circlular Base.

The Slant Height is the length of the diagonal slant of the cone from the top to the edge of the base.

The Height of the cone is the length from the top to the center of the circular base.

Cones

179

Surface Area = Area of the Base + Lateral AreaLateral Area= ½P ℓS.A. = B + ½P ℓ ℓ = Slant HeightP = Perimeter of Circular BaseB = Area of Circular BaseBecause the base is a circle.P = Circumference = 2πrL.A. = ½(2πr) ℓ = πr ℓ S.A. = πr2 + πr ℓ

Finding the Surface Area of a Right Cone

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Lateral SurfaceSlant Height ℓ

Base


180

LA = πr ℓ = π(6)(8)LA = 48π units2

LA = 150.80 units2

SA = πr2 + πr ℓ = π(6)2 + π(6)(8) = 36π + 48πSA = 84π units2

SA = 263.89 units2

Example: Find the lateral area and surface area of the right cone.

= 8

r = 6

ℓ

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181


h = 8

C = 12π unitsC = 2πr12π = 2πr 2π 2π6 units = r

62 + 82 = ℓ2

36 + 64 = ℓ2

100 = ℓ2

10 units = ℓ

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182


h = 8

C = 12π units

SA = πr2 + πr ℓ = π(6)2 + π(6)(10) = 36π + 60πSA = 96π units2

SA = 301.59 units2

LA = πr ℓ = π(6)(10)

LA = 60π units2

LA = 188.50 units2

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183

72 Find the lateral area of the right cone, in square units.

r = 4

ℓ = 9

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184

r = 4

ℓ = 9

73 Find the surface area of the right cone, in square units.

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185

74 Find the lateral area of the right cone, in square units.

h = 9

Base Area = 16π units2

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186

75 Find the surface area of the right cone, in square units.

h = 9

Base Area = 16π units2

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187

76 Find the length of the radius of the right cone if the lateral area is 50π units2?

ℓ = 10 Ans

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188

ℓ = 10

77 Find the height of the right cone if the lateral area is 50π units2?

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189

78 Find the slant height of the right cone if the surface area is 45π units2 and the diameter of the base is 6 units?

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190

79 Find the height of the right cone if the surface area is 45π units2 and the diameter of the base is 6 units?

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191

80 The Department of Transportation keeps 4 piles of road salt for snowy days. Each conical shaped pile is 20 feet high and 30 feet across at the base. During the summer the piles are covered with tarps to prevent erosion. How much tarp is needed to cover the conical shaped piles so that no part of them are exposed?

Ans

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LA = πr ℓ = π(15)(25)LA = 375π ft2LA = 1,178.10 ft2

192

Volume of a Prism


193

The volume of a solid is the amount of cubic units that a solid can hold.

Where area used square units, volume will use cubic units.

Prisms

194

Base

height

Base

ℓw

HV = BH

Specific PrismsRectangular Prism:

V = ℓwHCube: V = s3

Finding the Volume of a Prism

Prisms

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To avoid confusion with the "heights" when calculating the volume of a triangular prism, the height of the prism has been assigned "H". The triangular height will be "h" starting on slide #190

195

Does a prism need to be a right prism for the volume formula to work?

Think of a ream of paper

Stacked nicely it has 500 sheets.

If the stack is fanned, it still has 500 sheets.

So the volume doesn't change if the prism, stack of paper, is right or oblique.The formula V = BH works for all prisms.

Prisms

196

Example: Find the volume of the rectangular prism with a length of 2, a width of 6, and a height of 5.

V = ℓ w HV = 2(6)(5)V = 60 units3

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197

Example: The volume of a box is 48 ft3. If the height is 4 ft and width is 6 ft, what is the length?

V = ℓ w H48 = ℓ(6)(4)48 = 24 ℓ 24 242 ft = ℓ

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198

Example: Find the volume of the prism shown below.

10

611

Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem.

62 + b2 = 102

36 + b2 = 100b2 = 64 b = 8 units

Prisms

Mat

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you do this mentally? (MP5)- Referring to the Pythagorean TripleCan you find a shortcut to solve the problem? How would the shortcut make the problem easier? (MP8)- Referring to the Pythagorean TripleWhat labels could you use? (MP6)

Next, calculate the area of your base, BB = (1/2)(8)(6) = 24 units2

Finally, calculate your Volume.V = BHV = 24(11)V = 264 units3

199

Example: The volume of a cube is 64 m3, what is area of one face?

V = s3

64 = s3

4 m = s

Area of one faceA = 4(4)A = 16 m2

Prisms

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4 in

7 in

30°4 in.

x in.

Because the base is a regular polygon, we need to calculate the side length. To begin, figure out the central angle & top angle in the triangle.



360 6

60 2

Click

Click

Click

Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in.

Prisms

Teac

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201

4 in

7 in


Then, calculate the side length of your base.s = 2(2.31) = 4.62 in

Next, find the value of x using trigonometry, or special right triangles (if it applies).

tan 30 =

4tan30 = x x = 4√3 = 2.31 in. 3

x 4

Prisms

30°4 in.

x in.

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4 in

7 in


Next, use your value of s to find the Perimeter of your baseP = 6(4.62) = 27.72 in

Prisms

Click

Then, calculate the area of your base, B B = (1/2)aP = (1/2)(2.31)(27.72) = 32.02 in2

Finally, calculate your Volume.V = BhV = 32.02(7)V = 224.14 in3

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81 What is the volume of a rectangular prism with edges of 4, 5, and 7?

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82 What is the volume of a cube with edges of 5 units?

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83 If the volume of a rectangular prism is 64 u3 and has height 8 and width 4, what is the length?

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84 If a cube has volume 27 u3, what is the cubes surface area?

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85 Find the volume of the prism.

15

1220

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a2 + 122 = 152

a = 9 unitsB = (1/2)(12)(9) = 54 units2

V = 54(20)V = 1080 units3

208


7

2

6

6

6

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8

11


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need to start by finding the apothem = 4√3...30-60-90 triangleP = 6(8) = 48B = (1/2)(4√3)(48) = 96√3V = (96√3)(11)V = 1056√3 units3 V = 1829.05 units3

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88 A high school has a pool that is 25 yards in length, 60 feet in width, and contains the depth dimensions shown in the figure below.

If one cubic yard is about 201.974 gallons, how much water is required to fill the pool?

Shallow end

Deep end

3 ft9 ft

2 yds 4 yds19 yds

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V = 1(2)(20) + 4(3)(20) + 1/2(19)(1 + 3)(60) V = 40 + 240 + 2,280V = 2,560 cubic yardsWater:2,560(201.974) = 517,053.44 gallons of water

211

Volume of a Cylinder


212

base

base

height

r

r

Finding the Volume of a Cylinder

V = BhV = πr2h

213

Example: Find the volume of the cylinder with a radius of 4 and a height of 11.

V = π(4)2 (11)V = 176π units3

V = 552.92 units3

Cylinders

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Example: The surface area of a cylinder is 96π units2, and its radius is 4 units. What is the volume?

V = π(4)2 (8)V = 128π units3

V = 402.12 units3

SA = 2πr2 + 2πrh96π = 2π(4)2 + 2π(4)h96π = 32π + 8πh-32π -32π 64π = 8πh 8π 8πh = 8 units

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89 Find the volume of the cylinder with radius 6 and height 8.

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90 Find the volume of the cylinder with a circumference of 18π units and a height of 6.

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r = 8

h

91 Find the volume of the cylinder with a surface area of 653.12 u2 & a radius of 8 units. Use 3.14 as your value of π.

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218

92 The volume of a cylinder is 108π u3, and the height is 12 units. What is the surface area?

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219

93 The height of a cylinder doubles, what happens to the volume?

A Doubles

B Quadruples

C Depends on the cylinder

D Cannot be determined

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220

94 The radius of a cylinder doubles, what happens to the volume?

A Doubles

B Quadruples

C Depends on the cylinder

D Cannot be determined

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24"

4"

3"

95 A 3" hole is drilled through a solid cylinder with a diameter of 4" forming a tube. What is the volume of the tube?

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Volume of a Pyramid


223

Finding the Volume of a Pyramid

V = 1/3 BhSquare Base (B)

Slant Height ( )


ℓ

224

Example: Find the volume of the pyramid.

54

6

V = 1/3 BhB = 5(4) = 20h = 6 unitsV = 1/3 (20)(6)V = 40 units3

Volume of Pyramids

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88

5

88

5

4

h

click for extra diagram

Volume of Pyramids

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96 Find the volume of the pyramid.

76

5

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66

8

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12

12

10

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a


Regular Pentagonal PyramidFirst, find the side length of the regular pentagon using the apothem & trigonometric ratios.

= 72° = central

= 36° = top angle of the .72 2

360 5

tan36 =

x = 4tan36 = 2.91

Therefore, s = 2(2.91) = 5.82

x 4

Volume of Pyramids

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Example: Find the volume of the pyramid.Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pyth. Theorem.

8

2.91

ℓ

2.912 + ℓ 2 = 82

8.4681 + ℓ 2 = 64 ℓ 2 = 55.5319 ℓ = 7.45

Then, use the slant height & apothem w/ Pyth. Theorem to find the height.

7.45

4

h

Clic

k

42 + h2 = 7.452

16 + h2 = 55.5319h2 = 39.5319h = 6.29

Volume of Pyramids

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)How is Pythagorean Theorem used to calculate the segment lengths of pyramids? (MP7)

231


Last, find the Area of your Base & Volume.

B = 1/2 aPB = 1/2 (4)(29.1)B = 58.2 units2

V = 1/3 BhV = 1/3 (58.2)(6.29)V = 122.03 units3

Volume of Pyramids

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99 Find the volume of the right pyramid.

a

a = 5h = 12


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100 Find the volume of the right pyramid.

8

11


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B = 1/2(6.93)(48)B = 166.32 units2

234

A truncated pyramid is a pyramid with its top cutoff parallel to its base.

Find the volume of the truncated pyramid shown.

22

66

9

3Vtruncated = Vbig - Vsmall

Bbig = 6(6) = 36hbig = 3 + 9 = 12Vbig = 1/3 (36)(12)Vbig = 144 units3

Bsmall = 2(2) = 4hsmall = 3Vsmall = 1/3 (4)(3)Vsmall = 4 units3

Vtruncated = 144 - 4 Vtruncated = 140 units3

Volume of Pyramids

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101 Find the volume of the truncated pyramid.

22

8

8

12

3

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102 The table shows the approximate measurements of the Red Pyramid in Egypt and the Great Pyramid of Cholula in Mexico.

Approximately, what is the difference between the volume of the Red Pyramid and the volume of the Great Pyramid of Cholula?

A 6,132,867 cubic meters

B 4,455,000 cubic meters

C 2,777,133 cubic meters

D 1,677,867 cubic meters

Length Width HeightRed Pyramid 220 m 220m 104 mGreat Pyramid of Cholula 450 m 450 m 66 m

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103 Salt water comes in cylindrical containers that measure 10 feet high and have a diameter of 8 feet. Determine the height of the aquarium that should be used in the design. Show that your design will be able to store at least 3 cylindrical containers of water. When you finish, enter your value for h1 into your SMART Responder.

The Geometryville Aquarium is building a new tank space for coral reef fish shown in the figure below. The laws say that the dimensions of the tank must have a maximum length of 14 feet, a maximum width of 10 feet and a maximum height of 16 feet.

w

h1

h2

ℓ

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238

Volume of a Cone


239

r

height

Slant Height ℓ

Finding the Volume of a Cone

V = 1/3 Bh

V = 1/3πr2 h

240

Example: Find the volume of the cone.

9

7

V = 1/3 πr2 hV = 1/3 π(7)2 (9)V = 147π units3

V = 461.81 units3

Volume of a Cone

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Example: Find the volume of the cone.

12

4

V = 1/3 πr2 hV = 1/3 π(4)2 (8.94)V = 47.68π units3

V = 149.79 units3

r = 4, so d = 8With the right triangle, use Pythagorean Theorem to find the height of the pyramid.h2 + 82 = 122

h2 + 64 = 144h2 = 80, h = √80 = 8.94

Volume of a Cone

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Example: Find the volume of the cone, with lateral area of 15π units2 and a slant height 5 units.

LA = πr ℓ 15π = πr(5)15π = 5πr 5π 5π3 units = r

1) You know the Lateral area & slant height, so use the Lateral Area formula to calculate the radius.

Volume of a Cone

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h2 + 32 = 52

h2 + 9 = 25h2 = 16h = 4Note: 3-4-5 Pyth. Triple

2) Next, use the slant height & radius to calculate the height of the cone using Pythagorean Theorem.

Volume of a Cone

Click

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Would it help to draw a picture? (MP4 & MP5)How is Pythagorean Theorem used to calculate the segment lengths of cones? (MP7)

244

V = 1/3 πr2 h

V = 1/3 π(3)2 (4)V = 12π units3

V = 37.70 units3

3) Last, calculate the volume of the cone.

Volume of a Cone

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Mat

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you calculate your answer mentally? (MP5)What labels could you use? (MP6)

245

104 What is the volume of the cone?

8

d = 10

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r = 4

= 9ℓ

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247


1040°

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107 What is the volume of the truncated cone?

r = 8

r = 4

6

6

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249

Surface Area & Volume of Spheres


250

Recall the Definition of a Circle

The locus of points in a plane that are the same distance from a point called the center of the circle.

X

Y

Every point on the above circle is the same distance from the origin in the x, y plane.

Y

X

Spheres

251

The locus of points in space that are the same distance from a point.

Y

X

Z

Every point on the sphere above on the left side, is the same distance from the origin in space, the x, y, z plane.

X

Y

Y

X

Spheres

252

Y

X

Z

The Great Circle of a sphere is found at the intersection of a plane and a sphere when the plane contains the center of the sphere.

Spheres

253

Y

X

ZGreat Circles

Each of these planes intersects the sphere, and the plane contains the center of the sphere

254

InternationalDate Line

Great Circles The Earth has 2 Great Circles: Can you name them?

Click to reveal picture

The Equator

The Prime Meridian w/ the International Date Line

click

click

255

Great Circle

The Great Circle separates the Sphere into two equal halves at the center of the sphere.

256

Each half is called a Hemisphere

257

Cross Sections

A Cross Section is found by the intersection of a plane and a solid.

Cross - Section

(Click the top hemisphere to see the cross section.)

258

.small circles great

circle

The farther the cross section of the sphere is taken from its center the smaller the circle.

Cross Sections

259

82 8

r

Example: Find the radius of the cross section of the sphere that has a radius of 8 if the cross section is 2 from the center.

22 + r2 = 82

4 + r2 = 64r2 = 60r = √60 = 2√15 = 7.75

Cross Sections

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4

Example: A cross section of a sphere is 4 units from the center of the sphere and has an area of 16π units2. What is area of the great circle? Leave your answer in terms of π.

16π = πr2

r = 4 units in the cross section42 + 42 = r2 32 = r2

r =√32 = 4√2 = 2.83 = radius of sphereA = π(√32)2

A = 32π units2

Cross Sections

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Questions to help address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What tools do you need? (MP5)Can you calculate the answer mentally? (MP5)How is Pythagorean Theorem used to calculate the segment lengths of spheres? (MP7)

261

108 What is the area of the cross section of a sphere that is 6 units from the center of the sphere if the sphere has radius 8 units?

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262

109 What is the area of the great circle if a cross section that is 3 from the center has a circumference of 10π?

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263

110 The circumference of the great circle of a sphere is 12π units and a cross section has a circumference of 8π units. How far is the cross section from the center?

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264

rS.A. = 4πr2

Finding the Surface Area of the Sphere

Why is there no formula for lateral area?

A sphere doesn't have any bases, so the lateral area is the same as the surface area.

Click to reveal

265

r V = πr3 4 3

Finding the Volume of the Sphere

266

Example: Find the surface area & volume of a sphere with radius of 6 ft.

SA = 4π(6)2

SA = 144π units2

SA = 452.39 units2

V = π(6)3

V = 288π units3 V = 904.78 units3

4 3


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Example: Find the surface area & volume of a sphere that a great circle with area 24π units2?

SA = 4π(4.9)2

SA = 96.04π units2

SA = 301.72 units2

V = π(4.9)3

V = 156.87π units3 V = 492.81 units3

4 3

24π = πr2

π πr2 = 24r = 4.90 units


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Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is the surface area & volume of the sphere?

Radius of Cross Section36π = πr2

π πr2 = 36r = 6 units

Radius of Sphere102 + 62 = R2 136 = R2

R = √136 = 11.66 units


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SA = 4π(√136)2

SA = 544π units2

SA = 1,709.03 units2

V = π(√136)3

V = 2,114.69π units3 V = 6,643.50 units3

4 3

Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is the surface area & volume of the sphere?


click

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270

111 Find the surface area of a sphere with radius 10.

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271

112 Find the volume of a sphere with radius 10.

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272

113 What is the surface area of a sphere if a cross section 7 units from the center has an area of 50.26 units2?

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273

114 What is the volume of a sphere if a cross section 7 units from the center has an area of 50.26 units2?

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274

115 The volume of a sphere is 24π units3. What is the area of a great circle of the sphere?

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275

116 A recipe calls for half of an orange. Shelly use an orange that has a diameter of 3 inches. She wraps the remaining half of orange in plastic wrap. What is the amount of area that Shelly has to cover?

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276

Cavalieri's Principle


277


If two solids are the same height, and the area of their cross sections are equal, then the two solids will have the same volume.

278

1414 14

Which solid has the greatest volume?

224π703.72

None: All of the solids have the same volume.Click


2π8

4π

44

224π703.72

224π703.72Click Click Click

279

Example: A sphere is submerged in a cylinder. Both solids have a radius of 4. What is the volume of the cylinder not occupied by the sphere?

volume of cylinder - volume of sphere


π(4)2 (8) - 4/3 π(4)3

128π - 256/3 π 128/3 π units3 Click

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280

The result shows that the left over volume is equal to what other solid?

cone

According to Cavalieri, what can be said about the cross section? The cross section of the great circle of the sphere is equal to the circle cross section of the cylinder. Click


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Example:What is the radius of a sphere made from the cylinder of modeling clay shown?

If you are using clay to model both solids, what measurement is the same? Volume

15

5


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Question on this slide address MP standards:MP7

282

Therefore, calculate the volume of the cylinder first.


V = π(5)2 (15)V = 375π units3

Then create an equation to represent the problem and solve for r.

375π = 4/3 πr3

375 = 4/3 r3

281.25 = r3

r = 6.55 units

Click

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Questions on this slide address MP standards:"Then create an equation to represent...": MP2

Additional Questions to address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What connections do you see between these two solids? (MP4)What labels could you use? (MP6)

15

5

283

117 These 2 solids have the same volume, find the value of x.

11

r = 6

11

x 9

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284

118 These 2 solids have the same volume, find the value of x.

12

x

12

10

8

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285

Prism C

B = 20 in2

x

Prism D

B = 20 in2

y

Two prisms each with a base area of 20 square inches are shown.

Which statements about prisms C and D are true. Select all that apply. (Statements are on the next slide.)

286

119 Two prisms each with a base area of 20 square inches are shown.Which statements about prisms C and D are true. Select all that apply.

A If x > y, the area of a vertical cross section of prism C is greater than the area of a vertical cross section of prism D.

B If x > y, the area of a vertical cross section of prism C is equal to the area of a vertical cross section of prism D.

C If x > y, the area of a vertical cross section of prism C is less than the area of a vertical cross section of prism D.

D If x = y, the volume of prism C is greater than the volume of prism D, because prism C is a right prism.

E If x = y, the volume of prism C is equal to the volume of prism D because the prisms have the same base area.

F If x = y, the volume of prism C is less than the volume of prism D because prism D is an oblique prism.

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Similar Solids


288

Corresponding sides of similar figures are similar.

The prisms shown are similar. Find the values of x and y.

4

x2 6

9y

4 6 = 4

6 =

Similar Solids

x 9

36 = 6x 6 6 6 = x

4y = 12 4 4 y = 3

2 y

Click Click

Click

Click

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Mat

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Additional Questions to address MP standards:What information are you given? (MP1)What is the problem asking? (MP1)How can you represent the problem with symbols and numbers? (MP2)What connections do you see between these two solids? (MP4)What do you know about similar polygons that can be applied to this situation? (MP7)What labels could you use? (MP6)Can you find a shortcut to solve this problem? How would your shortcut make the problem easier? (MP8)- multiply corresponding side length by 2/3 (if given in the bigger prism) or 3/2 (if given in the smaller prism)

289

4

x2 6

9y

The ratio of similarity, k, is the common value that is multiplied to preimage to get to the image.

- Hint: it's the ratio of image : preimage

If the smaller prism is the preimage, then the value of k is

If the larger prism is the preimage, then the value of k is

click for the hint

Similar Solids

3/2

2/3

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290

120 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of x.

8

8

16

h2

x

y

3

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291

121 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of y.

8

8

16

h 2

x

y

3

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292

122 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of h.

8

8

16

h 2

x

y

3

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293

4

62

6

93

Consider the example of the prisms from earlier. The ratio of similarity from the smaller solid to the larger is 2:3.

Calculate the surface area of both solids. How do they compare?SAsmall = 2(6)(2) + 16(4) = 88 units2 SAbig = 2(3)(9) + 24(6) = 198 units2 SA Similarity ratio = 88:198 = 4:9 = 22:32

How do their volumes compare?Vsmall = 2(4)(6) = 48 units3 Vbig = 6(3)(9) = 162 units3 V Similarity ratio = 48:162 = 8:27 = 23:33

Similar Solids

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Click Click Click

Click Click Click

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Questions on this slide address MP standards:MP1, MP7 & MP8

294

Comparing Similar Figures

length in image

length in preimage= k

area in image

area in preimage = k2

volume in image

volume in preimage= k3

Tip

295

How many times bigger is the surface area of the sphere to the right?

How many times bigger is the volume of the sphere to the right?

r = 3

r = 9

Example:

How many times bigger is the radius of the sphere to the right?3 times bigger

9 times bigger

27 times bigger

Comparing Similar Figures

Click

Click

Click

SAsmall = 4π(3)2 = 36π units2

SAbig = 4π(9)2 = 324π units2

Vsmall = 4/3 π(3)3 = 36π units3

Vbig = 4/3 π(9)3 = 972π units3

Click

Click

Click

Click

Click

Click

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Questions on this slide address MP standards:MP1, MP7 & MP8

296

123 The scale factor of 2 similar pyramids is 4. If the surface area of the larger one is 64 units2, what is surface area of the smaller one?

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297

124 The scale factor of 2 similar right square pyramids is 3. If the area of the base of the larger one is 36 u2 and its height is 12, what is the volume of the smaller one?

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298

125 An architect builds a scale model of a home using a scale of 2 in to 5 ft. Given the view of the roof of the model, how much roofing material is needed for the house?

12 in

6 in8 in

5 in 4 in3 in

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PARCC Sample Questions

The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions.

Good Luck!


300

Question 6/11Daniel buys a block of clay for an art project. The block is shaped like a cube with edge lengths of 10 inches.

Daniel decides to cut the block of clay into two pieces. He places a wire across the diagonal of one face of the cube, as shown in the figure. Then he pulls the wire straight back to create two congruent chunks of clay.

PARCC Released Question - PBA - Calculator Section

Topic: Intro to 3-D Solids

301

126 Part A - Question #1: Daniel wants to keep one chunk of clay for later use. To keep that chunk from drying out, he wants to place a piece of plastic sheeting on the surface he exposed when he cut through the cube. Determine the newly exposed two-dimensional cross section.

A TriangleB ParallelogramC RectangleD RhombusE Square

Question 6/11



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127 Part A - Question #2:Daniel wants to keep one chunk of clay for later use. To keep that chunk from drying out, he wants to place a piece of plastic sheeting on the surface he exposed when he cut through the cube. Find the area of this newly exposed two-dimensional cross section. Round your answer to the nearest whole square inch.

Question 6/11



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128 Part B:Daniel wants to reshape the other chunk of clay to make a set of clay spheres. He wants each sphere to have a diameter of 4 inches. Find the maximum number of spheres that Daniel can make from the chunk of clay. Show your work.

Question 6/11


Topic: Cavaleri's Principle

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Question 10/11The Farmer Supply is building a storage building for fertilizer that has a cylindrical base and a cone-shaped top. The county laws say that the storage building must have a maximum width of 8 feet and a maximum height of 14 feet.

Topics: Volume of a Prism, Volume of a Cylinder, and Volume of a Cone


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129 Dump Trucks deliver fertilizer in loads that are 4 feet tall, 6 feet wide & 12 feet long. Farmer Supply wants to be able to store 2 dump-truck loads of fertilizer.Determine the height of the cylinder, h1, and a height of the cone, h2, that Farmer Supply should use in the design. Show that your design will be able to store at least two dump-truck loads of fertilizer. When you finish, enter your value for h1 into your Responder.

Question 10/11 Topics: Volume of a Prism, Volume of a Cylinder, and Volume of a Cone

PARCC Released Question - PBA - Calculator Section - SMART Response Format

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130 A rectangle will be rotated 360º about a line which contains the point of intersection of its diagonals and is parallel to a side. What three-dimensional shape will be created as a result of the rotation?

A a cube

B a rectangular prism

C a cylinder

D a sphere

Question 4/7

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PARCC Released Question - EOY - Non-Calculator Section


307

131 The table shows the approximate measurements of the Great Pyramid of Giza in Egypt and the Pyramid of Kukulcan in Mexico.

Approximately, what is the difference between the volume of the Great Pyramid of Giza and the volume of the Pyramid of Kukulcan?

A 1,945,000 cubic meters

B 2,562,000 cubic meters

C 5,835,000 cubic meters

D 7,686,000 cubic meters

PARCC Released Question - EOY - Calculator Section

Topic: Volume of a PyramidQuestion 8/25

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Question 11/25Two cylinders each with a height of 50 inches are shown.

Topic: Cavaleri's Principle


309

132 Which statements about cylinders P and S are true? Select all that apply.A If x = y, the volume of cylinder P is greater than the volume

of cylinder S, because cylinder P is a right cylinder.B If x = y, the volume of cylinder P is equal to the volume of

cylinder S, because the cylindres are the same height.C If x = y, the volume of cylinder P is less than the volume of

cylinder S, because cylinder S is slanted.D If x < y, the area of a horizontal cross section of cylinder P is

greater than the area of a horizontal cross section of cylinder S.

E If x < y, the area of a horizontal cross section of cylinder P is equal to the area of a horizontal cross section of cylinder S.

F If x < y, the area of a horizontal cross section of cylinder P is less than the area of a hoizontal cross section of cylinder S.

Question 11/25 Topic: Cavaleri's Principle


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133 Part AThe outer surface of the pipe is coated with protective material. How many square feet is the outer surface of the pipe? Give your answer to the nearest integer.

A steel pipe in the shape of a right circular cylinder is used for drainage under a road. The length of the pipe is 12 feet and its diameter is 36 inches. The pipe is open at both ends.

Question 13/25 Topic: Surface Area of a Cylinder


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134 Part BA wire screen in the shape of a square is attached at one end of the pipe to allow water to flow through but to keep people from wandering into the pipe. The length of the diagonals of the screen are equal to the diameter of the pipe. The figure represents the placement of the screen at the end of the pipe.

A 72 B 102 C 125

D 324 E 648 F 1,018and the area of the screen is ________ square inches.

Question 13/25 Topic: Surface Area of a Cylinder


The perimeter of the screen is approximately ________ inches,

Select from each set of answers to correctly complete the sentence.

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