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GEOMETRY CHAPTER 9

Name:_________Key _______________________ Date:______ Period:______

SECTION 9.6 THREE-DIMENSIONAL SHAPES

VOCABULARY

DEFINITION EXAMPLEPolyhedron/Polyhedra: a three-dimensional figure with many faces

Vertex/Vertices: vertices of the polygons are the vertices of the polyhe-dron

Faces: Each face of a polyhedron is a polygon

Edges: the borders of the faces that are also the line segments that join the vertices

Prism: a box shape where two of the faces, called bases, are parallel and congruent

Bases: in a prism, two faces that are parallel and congruent

Rectangular Prism: a prism whose bases are rectangles

Lateral Faces: Non-base faces of prisms or pyramids

Pyramid: has a polygon for a base and lateral faces that are triangles that meet at a point

Apex: the point where lateral faces of a pyramid meet

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DEFINITION EXAMPLECone: related to pyramids but with a circular base

Cylinder: related to a prism but with circular bases

Sphere: three-dimensional version of a circle, a figure formed by all points a fixed distance from a fixed point, called a center

Regular Shapes: each of its faces has equal sides and angles

Cube: a regular rectangular prism

Volume: measured by the number of unit cubes needed to fill it

Nets: created by cutting a shape along some of the edges and flattening it

Big Idea: How do you classify three-dimensional shapes? How do you find volumes of prisms?

A basic kind of three-dimensional figure is called a ____polyhedron__________. This word comes from comes from the Greek words poly, meaning ____many______, and hedra, meaning ______faces___________.

Each face of a polyhedron is a ________polygon_____________. The vertices of the polygon are the _______________vertices of the polygon____________________________________. The edges are the ______edges (borders)________ of the _______faces_________ that are also the _____line segments_________________ that join the vertices. Look at some examples below.

faceed

ge

vertex

vertex

face

edge

face

edge

vertex

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A box shape is the most common type of polyhedron called a ____prism_____________. In this shape, two of the faces, called ______bases____________, are parallel and congruent. _____Prisms____ are named by their _____bases____________. In the case of a box, the polyhedron is a ___________rectangular prism_______________, because the bases are _____rectangles___________. The faces that connect the two bases are parallelograms, and in this case rectangles. They are called _______lateral faces__________________.

EXPLORATION 1: PRISMS

As noted above, prisms are named by their bases. Rectangular prisms have rectangles as their bases. Although the faces of prisms are not always rectangles, in rectangular prisms, all of the faces are rectangles. Look at the rectangular prisms shown below:

You will need five different colored pencils, pens, or highlighters. Choose one color to draw the vertices on each prism. Count the vertices and record your information in the table.Using a different color, draw a line over the edges. Count and record the number of each in the table.You probably can guess what you should do next. Continue to choose a different color to shade and count the faces, bases, and lateral faces. Record your information in the table.

Figure Faces Edges Bases Lateral Faces VerticesA 6 12 2 4 8

B 6 12 2 4 8

C 6 12 2 4 8

D 6 12 2 4 8

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Pictured below are three different prisms:

___Rectangle_______ __Triangle________ ___Hexagon_________

___Rectangle_______ __Rectangle________ ___Rectangle_________

___Rectangular prism___ __Triangular Prism____ ___Hexagonal Prism____

On the lines below each shape state what shapes are the base faces, what shape are the lateral faces, and name each prism. Repeat the same process as before to fill out the table below.

Figure Faces Edges Bases Lateral Faces VerticesE 6 12 2 4 8

F 5 9 2 3 6

G 8 18 2 6 12

EXPLORATION 2: PYRAMIDS

Pyramids are another type of _____polyhedron______. Unlike a prism, pyramids only have one ______base________, which is a ________polygon___________, and triangular faces that meet at a point called the ___apex______. Like prisms, pyramids are named by their _____bases________. Identify the names of each pyramid below:

___Triangular Pyramid____ ___Rectangular Pyramid___ ___Pentagonal Pyramid___

EF G

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Compare and contrast prisms and pyramids on the lines below:

______(Student answers will vary) Pyramids have only one base while prisms have two, both are named by the shap of their bases, pyramids have triangular lateral faces and prisms have rectangular lateral faces. _______

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EXPLORATION 3: CONES, CYLINDERS AND SPHERES

Other common three-dimensional shapes include cones, cylinders and spheres.

Notice that _____cones__________ are related to pyramids, but have a ______circular______________ base. A

_______cylinder____________ in a similar way is related to a prism, but has __________circular_____________

bases. A ______sphere__________ is a three dimensional version of a circle, a figure formed by all points of a fixed

______distance__________ from a fixed point, called a _______radius (distance) or center (fixed point_________.

Cylinders Spheres Cones

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In the chart that follows, write some real-world examples of each figure.

3-D Figure Examples

Cylinder

Can of soup, glass of juice, water tower

Sphere

The sun, soccer ball, orange

Cone

Ice cream cone, party hat, roof of a round building

EXPLORATION 4: CUBES

A cube is a _______regular_________ rectangular prism because each of its _________faces_______ has

_______equal_________ sides and angles. All the cube’s faces are _________squares______________.

The cube is the simplest three-dimensional shape to measure having two parallel congruent square _________

bases___________ connected by four perpendicular congruent square lateral _______faces___________.

A cube one unit long, one unit wide, and one unit high has a volume of one _______cubic unit____________. Label the length, width, and height of the cube.

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Remember that each face of a three-dimensional shape is a ____polygon_________. In section 4, you learned about finding the area of a two-dimensional figure. If you were to cut along some of the edges and flatten the cube to create a net, it could look like the diagram pictured below.

Color two of the squares that would form the bases. Keep in mind the location of the squares when folded into a cube. Color the lateral faces. Figure the area of each square and write it inside each one. Add the area of each square together. This will give you the surface area of the cube. You will learn more about Surface Area in the part 2 book.

How many square units is the net? ______6 square units________

Can you think of other ways in which the cube can be cut into a net? Sketch your ideas using the graph paper given:

Student answers will vary.

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EXPLORATION 5: VOLUME

Using centimeter cubes, build a layer of length 2 cm and width 2 cm. Discuss the number of cubes used. What is the area of the figure base? _____4 square cm_________

Now add an identical layer on top of the base. How many total cubes are used? ____8 cubes_____ To find the answer, you can take the answer for the number of cubes used in the base and multiply it by the number of layers. We now have: _2__ · __2__ · _2__ = __8___ cubic centimeters. Common abbreviations for cubic centimeters are cu cm and cm3. This is the volume of the cube you constructed.

The ____volume______ of a three-dimensional shape is measured by the number of cubic units needed to fill it. Consider a cube that is 5 units long, 5 units wide, and 5 units tall. How many cubic units would be needed to fill this shape?

Look back at Exploration 4 where you labeled the length, width, and height of a cube that measured one unit for each dimension. Imagine placing those cubes inside the larger cube as pictured above. It would take __5__ of the unit cubes to make a lengthwise row. How many rows would you need to cover the surface? __5_____ How many rows would you have to stack to fill the cube to the top with the smaller unit cubes? __5____ The product of these dimensions would give us the volume of the cube: __5__ · __5__ · _5___ = ___125__ cubic units. Since we see a factor that is repeating, we could write this expression in exponential form as __53___ = __125_ cubic units.

What is the difference between the unit used to denote area and the unit used to denote volume?

Area is units squared. Volume is units cubed.

EXPLORATION 6: CONVERTING UNITS

How many cubic inches are there in one cubic foot? Begin this problem by creating a sketch of a cube and labeling its dimensions.

Did you label in feet or inches? ___inches____ Why? _________We want to find cubic inches, not feet.__________

_______________________________________________________________________________________

12 in

12 in

12 in

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The problem states that we have a cube that measures one cubic foot, but it is asking us how many cubic inches there

are in that one cubic foot. Write the inch to foot conversion: _____12 inches = 1 foot_________

We have found that the volume of a cube is found by multiplying side times side times side, s · s · s, or s3.

To start, write your formula using exponential notation, substituting 12 in. for s in the formula:

s3 = V

In expanded form, substitute 12 in. for s in the formula:

s · s · s = V

Now, write the formula. Multiply to find the product, which is the volume of the cube. Remember to write:

V = _____, using the appropriate units.

(Show your work here.)

123=V

12 · 12 · 12 = V

length · width ·height = V

(12 ·12)(12) = (144)(12) = 1728 cubic inches

EXPLORATION 7: VOLUME OF A RECTANGULAR PRISM

The rectangular prism below has edges that are 2, 3, and 4 units long.

How many unit cubes does it take to fill the box? The 2 x 3 base rectangles can be cut into __6_____ unit squares,

and the height is ____4 units_____. To find the volume, multiply the area of the base times the height: __________

(6)(4)=24_____________ . Therefore, V = ____24 cubic inches___________.

23

4

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FORMULA 9.6: Volume of a PrismThe volume formula for a prism can be written by volume = area of base · height or V = Bh with B = area of the base and h = height of the prism. In particular, for a rectangular prism, B = lw so V = lwh.

PROBLEMS:1. Fill in the table with the number of vertices, edges, faces, and a sketch of each shape.

Name of Shape: Vertices: Edges: Faces: Sketch:

Triangular Prism 6 9 5

Rectangular Prism 8 12 6

Pentagonal Prism 10 15 7

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2. Find the volume of a cube with sides of length of 6 cm.

(6)(6)(6) = (36)(6) = 216 cubic cm

3. What is the volume, in cubic feet, of a cube with side lengths of 6 inches? (Hint: Don’t forget to convert!)

6 in = .5 ft

(.5)(.5)(.5) = (.25)(.5) = .125 cubic ft

or 21

21

21 1

21 1

4 8= =a a a a a ak k k k k k cubic ft

V = ___.125 cubic ft or (1/8) cubic ft___

4. A koi pond has dimensions 8 feet x 6 feet x 3 feet. Determine the volume of the pond.

(8)(6)(3) = (48)(3) = 144 cubic ft

The pond has a volume of _________144 cubic ft______________.

5. If the volume of the form that makes a sugar cube holds 8 cm3, what is the measure of the side length?

(2)(2)(2) = 8 by factoring

______________2 cm_____________________________________________

6. Every rectangular box made by the Bow-Wow Box Company has a length of 2 feet. Find the volume of each box that the company makes.

Width (ft.): Height (ft.): Volume: (remember to use the appropriate unit)

1 4 8 cubic ft

2 6 24 cubic ft

3 8 48 cubic ft

4 10 80 cubic ft

6 cm

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7. Find the missing value, h, in the formula for the volume of a rectangular prism. Be sure to indicate the units for h.

12.5 m · 5 m · h = 187.5 m3

(62.5 sq. m)(h) = 187.5 cubic mh=(187.5)/(62.5)

h=3 m

h = ___3 meters________

SUMMARY (What I learned in this section)

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