Grade 8 Building the Cylinder Volume Formula: Graded Practice Set

Position in the topic

Use this record after the linked practice sets on cylinder-volume derivation and formula use:

• [rea.m08.geometry-measurement.volume.cylinder-volume-derivation-practice-set]
• [rea.m08.geometry-measurement.volume.cylinder-volume-formulas.practice-set]
• [rea.m08.geometry-measurement.volume.basic-formulas.formula-selection.practice-set]
• [rea.m08.geometry-measurement.volume.basic-formulas.practice-set]

This set is narrower: it concentrates on building the formula V = pi r^2 h from structure.

What this set practices

• seeing a cylinder as many equal circular layers
• identifying the base area as pi r^2
• expressing volume as base area x height
• distinguishing radius from diameter
• deciding when an answer should stay exact and when to approximate
• reverse reasoning about missing dimensions

Quick method reminder

Do not re-study the full lesson here; use the linked records for that. For this set, the working idea is:

1. Find the area of one circular base.
2. Multiply by the height.
3. Keep units cubic.

So for a cylinder, V = Bh = (pi r^2)h = pi r^2 h.

Difficulty tags

• D1 straightforward identification or substitution
• D2 multi-step setup with unit care
• D3 reverse or comparison reasoning
• D4 explanation or non-routine derivation reasoning

Problems

A. Building the formula from structure

1. D1 A cylinder is made of circular layers. Each layer has area 20 cm^2, and the height is 9 cm. What is the volume?

2. D1 A cylinder has base area 13 pi m^2 and height 4 m. Write its volume in exact form.

3. D1 A cylinder has radius 3 cm and height 8 cm. Build the volume formula step by step and find the exact volume.

4. D1 A cylinder has diameter 10 cm and height 7 cm. Build the formula carefully and find the exact volume.

5. D1 A cylinder has radius 1.5 m and height 12 m. Write the exact volume in terms of pi.

6. D2 The base of a cylinder has area 49 pi cm^2. What is the radius of the base?

7. D2 The base area of a cylinder is 81 pi mm^2, and the height is 11 mm. Find the exact volume.

8. D2 A cylinder has radius 6 cm. If its height doubles, how does the volume change?

B. From circles to cylinders

9. D2 A circular base has radius 4 cm. Find the base area, then use height 15 cm to find the cylinder volume in exact form.

10. D2 A cylinder has diameter 14 m and height 5 m. Find the exact volume.

11. D2 A soup can is modeled as a cylinder with radius 3.5 cm and height 10 cm. Find the volume to the nearest tenth of a cubic centimetre.

12. D2 A cylinder has radius 9 mm and volume 729 pi mm^3. Find its height.

13. D2 A cylinder has height 20 cm and exact volume 500 pi cm^3. Find its radius.

14. D3 Two cylinders have the same height. Cylinder A has radius 2 cm. Cylinder B has radius 5 cm. How many times the volume of A is the volume of B?

15. D3 Two cylinders have the same radius. Cylinder P has height 7 cm. Cylinder Q has height 21 cm. Compare their volumes.

16. D3 A cylinder has radius 4 cm and height 10 cm. Another has radius 8 cm and height 10 cm. Compare the volumes and explain briefly.

C. Reverse and reasoning problems

17. D3 A cylinder has exact volume 432 pi cm^3 and height 12 cm. Find the radius.

18. D3 A cylinder has exact volume 300 pi m^3 and radius 5 m. Find the height.

19. D3 A cylinder has diameter 16 cm and volume 768 pi cm^3. Find the height.

20. D4 A student says, “If the radius doubles, the volume doubles.” Give a counterexample using a fixed height of 6 cm, starting from radius 2 cm.

21. D4 A cylinder and a prism both have height 9 cm and base area 24 cm^2. Compare their volumes and explain what this shows about V = Bh.

22. D4 A cylinder has radius r and height h. Another has radius 2r and height h/2. Compare their volumes algebraically.

23. D4 A cylinder has exact volume 980 pi cm^3. One possible height is 5 cm. Find the corresponding radius. Then find a different whole-number height that gives a different rational radius.

24. D4 A cylindrical water container has inside diameter 12 cm and inside height 18 cm. It is filled to half its height. Find the amount of water in exact form and to the nearest tenth. Explain why halving the height halves the volume here.

Answer key

1. 180 cm^3
2. 52 pi m^3
3. 72 pi cm^3
4. 175 pi cm^3
5. 27 pi m^3
6. 7 cm
7. 891 pi mm^3
8. The volume doubles.
9. Base area 16 pi cm^2; volume 240 pi cm^3
10. 245 pi m^3
11. 384.8 cm^3 approximately
12. 9 mm
13. 5 cm
14. Cylinder B has 25/4 times the volume of Cylinder A.
15. Cylinder Q has 3 times the volume of Cylinder P.
16. The second cylinder has 4 times the volume.
17. 6 cm
18. 12 m
19. 12 cm
20. Original volume 24 pi cm^3; new volume 96 pi cm^3; volume becomes 4 times as large, not 2 times.
21. They have equal volumes: 216 cm^3 each. This shows volume depends on base area and height, not the base shape by itself.
22. The second volume is 2 times the first.
23. For height 5 cm, radius 14 cm. Example of another whole-number height: 20 cm, giving radius 7 cm.
24. Exact volume 648 pi cm^3; approximate volume 2035.8 cm^3; halving the height halves Bh because the base area stays constant.

Full solutions for the hardest third

17. Exact volume 432 pi cm^3, height 12 cm

Use V = pi r^2 h.

432 pi = pi r^2 (12)

Divide both sides by 12 pi:

r^2 = 432/12 = 36

r = sqrt(36) = 6

So the radius is 6 cm.

18. Exact volume 300 pi m^3, radius 5 m

Start with V = pi r^2 h.

300 pi = pi (5^2) h 300 pi = 25 pi h

Divide by 25 pi:

h = 300/25 = 12

So the height is 12 m.

19. Diameter 16 cm, volume 768 pi cm^3

The radius is half the diameter:

r = 16/2 = 8 cm

Now use V = pi r^2 h:

768 pi = pi (8^2) h 768 pi = 64 pi h

Divide by 64 pi:

h = 768/64 = 12

So the height is 12 cm.

20. Counterexample to “If the radius doubles, the volume doubles.”

Keep height fixed at 6 cm.

First cylinder: r = 2 cm

V = pi r^2 h = pi (2^2)(6) = 24 pi cm^3

Second cylinder: radius doubles to 4 cm

V = pi (4^2)(6) = 96 pi cm^3

Compare:

96 pi / 24 pi = 4

So doubling the radius makes the volume 4 times as large, not 2 times as large, when height stays the same. The reason is that radius is squared in pi r^2 h.

21. Cylinder and prism with the same base area and height

Both solids have:

• base area 24 cm^2
• height 9 cm

Use V = Bh for each solid:

V = 24 x 9 = 216 cm^3

So both volumes are 216 cm^3.

This shows that volume can be built from the same structure, base area x height, even when the base shapes are different. For a cylinder, the base happens to be a circle, so B = pi r^2.

22. Compare a cylinder with radius r, height h to one with radius 2r, height h/2

First cylinder:

V1 = pi r^2 h

Second cylinder:

V2 = pi (2r)^2 (h/2)

Simplify:

V2 = pi (4r^2)(h/2) V2 = 2 pi r^2 h

Since V1 = pi r^2 h,

V2 = 2V1

So the second cylinder has twice the volume of the first.

23. Exact volume 980 pi cm^3

First, use height 5 cm.

980 pi = pi r^2 (5)

Divide by 5 pi:

r^2 = 980/5 = 196

r = sqrt(196) = 14

So if the height is 5 cm, the radius is 14 cm.

Now find a different whole-number height that gives a different rational radius. We want 980/h to be a perfect square or at least a rational square. Choose h = 20.

Then:

r^2 = 980/20 = 49 r = 7

So another valid choice is height 20 cm with radius 7 cm.

24. Half-filled cylindrical container

Inside diameter 12 cm means radius 6 cm.

The full height is 18 cm, so half-filled means water height 9 cm.

Use V = pi r^2 h:

V = pi (6^2)(9) V = pi (36)(9) V = 324 pi

Check carefully: that is the volume for height 9 cm, and since 36 x 9 = 324, the water volume is 324 pi cm^3.

Now approximate:

324 pi ≈ 1017.9 cm^3

Why does halving the height halve the volume? Because for a cylinder with fixed radius, the base area pi r^2 stays constant, so volume is directly proportional to height:

V = (constant) x h

If height is cut in half, volume is cut in half.

Correction note for Problem 24

If the cylinder were filled to half its full volume, that would be 648 pi cm^3. But the problem states it is filled to half its height, so the correct water volume is 324 pi cm^3 ≈ 1017.9 cm^3.

Common mistakes and repairs

• Using diameter in place of radius. Repair: mark r = d/2 before substituting.

• Forgetting to square the radius. Repair: write the base area separately first: B = pi r^2.

• Writing square units instead of cubic units. Repair: area is square units; volume is cubic units.

• Thinking every dimension change affects volume the same way. Repair: height changes volume linearly, but radius changes volume quadratically.

• Converting exact answers to decimals too early. Repair: keep pi until the last step unless the question asks for an approximation.

Short self-check

You are ready to move on if you can do all of these without prompting:

• explain why V = Bh works for cylinders
• replace B with pi r^2
• switch correctly between radius and diameter
• solve for a missing height or radius from an exact volume
• explain how volume changes when radius or height changes