- 1 - Geometry Rules! Chapter 12 Notes

Notes #27: Surface Area and Volume of Rectangular Prisms and Cubes (Section 12.1) A. Let’s Draw! Rectangular Prism: - draw a rectangle, this is the front face of your prism - draw a congruent rectangle again behind and to the right of the first - connect the appropriate corners (bottom right to bottom right, etc.)

Cube: (A rectangular prism with ________________ _______________) - draw a square, this is the front face of your prism - draw a congruent square again behind and to the right of the first - connect the appropriate corners (bottom right to bottom right, etc.)

B. Key Vocabulary Prism: A 3D solid with 2 parallel and congruent _____________ and ____________________ faces Base: Altitude: Lateral Faces: Lateral Edges:

- 2 -

Lateral Area of a Prism

LA = __________________________________

Total Area of a Prism

_____________________________________

TA = LA + 2B

TA = (______________) + 2(____________)

Volume of a Prism

________________________________________

V = Bh V = (_____________)(_____________)

Total Area of a Cube

TA = 6e2

Volume of a Cube

V = e3

- 3 - For #1-3, Complete the table referring to rectangular prisms:

1.) 2.) 3.) For #4-5, Complete the table referring to cubes with side length e:

4. 5. e 2 cm

TA 600 in2 V

4.) 5.)

1. 2. 3. l 5 cm 2 in 3 m w 4 cm 2 m h 3 cm 6 in

LA 40 cm TA V 60 in3

- 4 - Notes #28: Surface Area and Volume of Prisms (Section 12.1) Prism with bases that are equilateral triangles

Prism with bases that are isosceles triangles

Prism with bases that are right triangles

Prism with bases that are isosceles trapezoids

Prism with bases that are regular hexagons

Prism with bases that are rhombuses

For #1-5, refer to the right prism shown: 1. The prism is called a right _______________ prism. 2. How many lateral faces are there? ______ How many bases? ____ 3. What kind of shape is each lateral face?______________________ 4. Name a lateral edge: ______ Name an altitude: ______ 5. If the bases are regular hexagons with edges 8m, find: a) Lateral Area b) Base Area c) Total Area d) Volume

- 5 - For #6-10, refer to the right prism shown: 6. The prism is called a right _______________ prism. 7. How many lateral faces are there? ______ How many bases? ____ 8. What kind of shape is each lateral face?______________________ 9. Name a lateral edge: ______ Name an altitude: ______ 10. If the bases are isosceles trapezoids with legs 5m and bases 6 and 14 find: a) Lateral Area b) Base Area c) Total Area d) Volume Find the lateral area, total area, and volume of each prism: 11.)

- 6 - 12.) A right prism with a base of a regular hexagon with perimeter 12m, h = 5m 13.) Base is an isosceles triangle with base 12cm and legs 10cm, h = 5cm Notes #29: Surface Area and Volume of Square Pyramids (Section 12.2) Let’s Draw: - draw a rhombus (sleepy square) - mark its center - from the center, draw a height - connect all corners of the rhombus to the top this height

- 7 -

Key Vocab Vertex: Base: Altitude: Lateral Edges: Lateral Faces: Slant Height: (l)

Key Right Triangles

D

A

C

B

X

E

E

X F

E

F C

F

4

3

Lateral Area of a Pyramid

__________________________________________

Or

2plLA =

LA = (__________________)(_______________)

2

Total Area of a Pyramid ________________________________

TA = LA + B

TA = (_______________) + (_____________)

- 8 -

4

3

Volume of a Pyramid

________________________________________

V = 3Bh

V = (_____________)(_____________) 3

For #1-6, refer to the regular square pyramid: 1. XF = 2. l = 3. Area of ∆EBC = 4. Lateral Area = 5. Total Area = 6. Volume =

D

A

C

B

X

E

F

Base edge = 10cmHeight = 12cm

For #7–8, Complete the table for regular square pyramids: (right triangles)

7. 8. height, h

3 6

slant height, l 5

base edge 16

lateral edge

lh

7.) 8.)

- 9 - Sketch each pyramid and find its lateral area, total area, and volume: 9. base edge = 8, height = 3 10. base edge = 12, slant height = 10 Complete: 11. A regular square pyramid has base area of 36m2 and a volume of 48m3. Find its height, base edge, lateral edge, lateral area, and total area. h = _____ base edge = _____ lateral edge = _____ LA = _____ TA = _____

- 10 - Notes #30: Surface Area and Volume of Cylinders and Cones (Section 12.3) Cylinders: Let’s Draw: - Draw a squished circle (ellipse) - Draw a congruent ellipse above it - connect the right and left “sides” with straight lines Key Pieces Base: Altitude: radius:

Lateral Area of a Cylinder

__________________________________ LA = 2 rhπ

LA =(_______________________)(___________)

Total Area of a Cylinder

_________________________________ TA = LA + 22 rπ

TA = (____________) + 2(_____________)

Volume of a Cylinder __________________________________

V = 2r hπ

V = (_____________)(_____________)

- 11 - Sketch each cylinder. Then find its lateral area, total area, and volume: 1. r = 6, h = 5 2. diameter = 10, h = 3 3. r = 2, h = 7 4. The volume of a cylinder is 27π . If r = h, find r. 5. The lateral area of a cylinder is 20π . If h = 5, find r.

- 12 - Cones Let’s Draw: - draw an ellipse (squished circle) - from its center, draw a height - draw two straight lines connecting the height to the ellipse Key Pieces base: altitude: radius: slant height: (l)

Lateral Area of a Cone

LA = rlπ

OR

LA = (_______________________)(___________) 2

Total Area of a Cone

TA = LA + 2rπ

TA = (_______________) + (_____________)

- 13 - Volume of a Cone

V = 2

3r hπ

V = (_____________)(_____________)

3

Sketch each cone. Then find each value: 6. r = 3, h = 4 Slant height =_____ LA = _____ TA = _____ V = _____

7. diameter = 12, h = 8

Slant height =_____

LA = _____

TA = _____

V = _____

8. h = 4, l = 10 radius =_____ LA = _____ TA = _____ V = _____

9. r = 8, LA = 96π

Slant height =_____

height = _____

TA = _____

V = _____

- 14 - Notes #31: Surface Area and Volume of Spheres (Section 12.4) Areas and Volumes of Similar Solids (Section 12.5) Spheres

Total Area of a Sphere

A = 24 rπ

Volume of a Sphere

V = 34

3rπ

Sketch each sphere. Then find the indicated values: 1. r = 6 2. diameter = 10 A = _______ A = _______ V = _______ V = _______

- 15 - 3. The area of a sphere is 64π . Find 4. The volume of a sphere is 500

3π . Find

the radius and volume of the sphere. The radius and area of the sphere. r = ________ r = ________ V = _________ V = ________ A plane passes h cm from the center of a sphere with radius r. Find the radius and area of the circle of intersection, shaded in the diagram, for the given values. 5. r = 5, h = 4 6. r = 15, h = 12

- 16 - Areas and Volumes of Similar Solids Similar Solids: Given the two similar solids, find: (a) the ratio of their side lengths (b) the scale factor (c) the ratio of their base areas (d) the ratio of their total volumes Complete: If the scale factor of two similar solids is a:b, then

• The ratio of corresponding lengths is ________ (examples of lengths: ) • The ratio of bases areas, lateral areas, or total areas is ________ • The ratio of volumes is ________

Complete the table below for two similar solids: 7. 8. 9. 10. Scale Factor 3:5 Ratio of base perimeters Ratio of heights 2:1 Ratio of lateral areas 4:9 Ratio of total areas Ratio of volumes 27:64

- 17 - Geometry: Chapter 12 Study Guide

Please include units with your answers and when appropriate, leave your answers in terms of π

Prisms: 1. Find the lateral area and total area of a rectangular prism with base dimensions of 4ft by 5ft, and prism height of 6ft. 3. The base of a prism is an equilateral triangle with sides 6m. If the height of the prism is 10m, find the lateral area, total area, and volume of the prism. 5. The total area of a cube is 150cm2. Find the length of an edge of the cube and its volume.

2. Find the volume of a rectangular prism with base dimensions of 2in by 7in and prism height of 5in. 4. The base of a prism is a regular hexagon with sides 4in. If the height of the prism is 8in, find the lateral area, total area, and volume of the prism. 6. The base of a prism is an isosceles trapezoid with bases 9 and 19 and legs of 13cm. If the height of the prism is 10cm, find its lateral area, total area, and volume.

Pyramids: (These are regular square pyramids) 7. If the height of a pyramid is 6ft and its slant height is 10ft, find the length of a base edge and the length of a lateral edge.

8. If the base edge of a pyramid is 14cm and its lateral edge is 25cm, find the slant height and height of the pyramid.

9. Find the lateral area, total area, and volume of the square pyramid with base edge of 12 and height of 8. 11. Find the lateral area, total area, and volume of the square pyramid with height of 12 and slant height of 13.

10. Find the lateral area, total area, and volume of the square pyramid with base edge of 8 and slant height of 5. 12. A square pyramid has a base area of 64ft2 and a volume of 192 ft3. Find its height.

Cylinders: 13. Find the lateral area, total area, and volume of the cylinder with radius 3m and height 8m.

14. Find the lateral area, total area, and volume of the cylinder with radius 3 2 cm and height 8cm.

- 18 - 15. The volume of a cylinder is 50π in3. If the height is 2in, find the radius, the lateral area, and the total area.

16. The lateral area of a cylinder is 18π km2. If the height is 6km, find the radius, the total area, and the volume.

Cones 17. Find the lateral area, total area, and volume of the cone with radius 4cm and height 3cm.

18. Find the lateral area, total area, and volume of the cone with radius 8m and slant height 17m.

19. Find the height, slant height, total area, and volume of a cone with radius 5ft and lateral area of 65π ft2.

20. Find the height, slant height, lateral area and total area of the cone with radius 6cm and volume 96π cm3.

Spheres 21. Find the area and volume of a sphere with radius 7m.

22. Find the area and volume of a sphere with diameter 12ft.

23. Find the radius and area of a sphere with volume 32

3π cm3.

24. A plane passes 3 cm from the center of a sphere with radius 5cm. Find the area of the circle of intersection.

Ratio of Dimensions 25. The scale factor of two similar solids is 2:5. What is the ratio of: a) their corresponding sides? b) the perimeters of their bases? c) their lateral areas? d) their volumes?

26. The scale factor of two similar solids is 3:4. The base area of the larger solid is 48m2. Find the base area of the smaller solid.

27. The ratio of the base areas of two similar solids is 4:9. What is the ratio of: a) their corresponding sides? b) their total areas? c) the perimeter of their bases? d) their volumes? Area Review 29. Find the area of a regular hexagon with apothem 3cm.

28. The ratio of the volumes of two similar solids is 1:8. What is the ratio of: a) their base perimeters? b) their lateral areas? c) their total areas? 30. Find the area of a regular hexagon with perimeter of 48 m.