Answer Keys: Volume of Cylinders, Cones, and Spheres

Student Concept Worksheet Answers

1

V = 144 pi cubic cm

V = pi (4)^2 (9) = pi (16)(9) = 144 pi cubic cm

2

radius = 4 in; V = 80 pi cubic in

Diameter 8, so radius = 4. V = (1/3) pi (4)^2 (15) = (1/3) pi (16)(15) = (1/3)(240 pi) = 80 pi cubic in

3

V = 36 pi cubic m

V = (4/3) pi (3)^3 = (4/3) pi (27) = 36 pi cubic m

4

Ratio: 1 : 3

Cone volume = (1/3) pi r^2 h; Cylinder volume = pi r^2 h. The cone is 1/3 of the cylinder, so the ratio is 1:3.

Practice Worksheet Answers

Part A: Volume of Cylinders

#	Answer	Work
1	300 pi cm^3	V = pi (5)^2 (12) = pi (25)(12) = 300 pi
2	490 pi in^3	V = pi (7)^2 (10) = pi (49)(10) = 490 pi
3	240 pi m^3	Diameter 8, radius = 4. V = pi (4)^2 (15) = pi (16)(15) = 240 pi
4	600 pi ft^3	Diameter 20, radius = 10. V = pi (10)^2 (6) = pi (100)(6) = 600 pi

Part B: Volume of Cones

#	Answer	Work
5	108 pi cm^3	V = (1/3) pi (6)^2 (9) = (1/3) pi (36)(9) = (1/3)(324 pi) = 108 pi
6	42 pi in^3	V = (1/3) pi (3)^2 (14) = (1/3) pi (9)(14) = (1/3)(126 pi) = 42 pi
7	100 pi m^3	Diameter 10, radius = 5. V = (1/3) pi (5)^2 (12) = (1/3)(300 pi) = 100 pi
8	294 pi ft^3	Diameter 14, radius = 7. V = (1/3) pi (7)^2 (18) = (1/3)(882 pi) = 294 pi

Part C: Volume of Spheres

#	Answer	Work
9	36 pi cm^3	V = (4/3) pi (3)^3 = (4/3) pi (27) = 36 pi
10	288 pi in^3	V = (4/3) pi (6)^3 = (4/3) pi (216) = 288 pi
11	(500/3) pi m^3	Diameter 10, radius = 5. V = (4/3) pi (5)^3 = (4/3) pi (125) = 500/3 pi (about 166.67 pi)
12	972 pi ft^3	Diameter 18, radius = 9. V = (4/3) pi (9)^3 = (4/3) pi (729) = 972 pi

Part D: Word Problems

13

24 pi m^3 = approximately 75.4 m^3

Diameter 4, radius = 2. V = pi (2)^2 (6) = pi (4)(6) = 24 pi. Using pi = 3.14: 24 x 3.14 = 75.36

14

(40/3) pi cm^3 = approximately 41.9 cm^3

V = (1/3) pi (2)^2 (10) = (1/3) pi (4)(10) = (40/3) pi. Using pi = 3.14: (40/3) x 3.14 = 41.87

15

Approximately 435 in^3

Diameter 9.4, radius = 4.7. V = (4/3) pi (4.7)^3 = (4/3) pi (103.823) = (4/3)(3.14)(103.823) = 435.01

16

(2048/3) pi ft^3 or approximately 682.67 pi ft^3

Cylinder: V = pi (8)^2 (20) = 1280 pi
Hemisphere (half sphere): V = (1/2)(4/3) pi (8)^3 = (2/3) pi (512) = (1024/3) pi
Total: 1280 pi + (1024/3) pi = (3840/3) pi + (1024/3) pi = (4864/3) pi = approximately 1621.33 pi ft^3

Challenge Questions:

#17: With h = 2r: Cylinder: V = pi r^2 (2r) = 2 pi r^3
Cone: V = (1/3) pi r^2 (2r) = (2/3) pi r^3
Sphere: V = (4/3) pi r^3
Comparing: Cylinder (2 pi r^3) > Sphere (4/3 pi r^3 = 1.33 pi r^3) > Cone (2/3 pi r^3 = 0.67 pi r^3)
The cylinder has the greatest volume.

#18: If you double the radius: Original V = (4/3) pi r^3. New V = (4/3) pi (2r)^3 = (4/3) pi (8r^3) = 8 x (4/3) pi r^3. The volume becomes 8 times greater because r is cubed, and 2^3 = 8.

FAST Practice Quiz Answers

1

C) 160 pi cubic cm

V = pi (4)^2 (10) = pi (16)(10) = 160 pi

2

A) 108 pi cubic inches

Diameter 12, radius = 6. V = (1/3) pi (6)^2 (9) = (1/3) pi (36)(9) = (1/3)(324 pi) = 108 pi

3

C) 288 pi cubic cm

V = (4/3) pi (6)^3 = (4/3) pi (216) = 288 pi

4

Select: A, C, E

A) Cone has 1/3 the volume of same cylinder - TRUE
B) Double radius = double volume - FALSE (volume quadruples because r is squared)
C) Sphere uses r cubed - TRUE
D) Same dimensions = same volume - FALSE (cone is 1/3 of cylinder)
E) Volume uses cubic units - TRUE

5

B) 226 cubic meters

Diameter 6, radius = 3. V = pi (3)^2 (8) = pi (9)(8) = 72 pi. Using 3.14: 72 x 3.14 = 226.08 = 226 cubic meters

6

A) 300 pi + (250/3) pi cubic feet

Cylinder: V = pi (5)^2 (12) = 300 pi
Hemisphere: V = (1/2)(4/3) pi (5)^3 = (2/3) pi (125) = (250/3) pi
Total: 300 pi + (250/3) pi cubic feet

FAST Quiz Scoring Guide

Score	Interpretation	Recommended Action
6/6	Mastery	Ready for composite figures and volume applications.
4-5/6	Approaching Mastery	Review specific formulas and watch for diameter vs radius errors.
2-3/6	Developing	Reteach formulas with visual models. Focus on one shape at a time.
0-1/6	Needs Intervention	Small group instruction with manipulatives. Start with cylinder formula only.

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