Layering a Cylinder: Graded Practice Set

Position in the topic

This record is the independent practice companion for rea.m08.geometry-measurement.volume.cylinder-derivation.layering. Learn or review the idea first in:

• rea.m08.geometry-measurement.volume.cylinder-derivation.layering for the unit overview
• rea.m08.geometry-measurement.volume.cylinder-derivation.layering-lesson for the lesson development
• rea.m08.geometry-measurement.volume.cylinder-derivation.layering-worked-examples for guided examples
• rea.m08.geometry-measurement.volume.cylinder-volume-derivation-practice-set for broader mixed derivation practice

This record does not reteach the full derivation. It focuses on practice using the layering model: a cylinder is many congruent circular layers stacked through a height, so

volume = area of one layer x number of layers = pi r^2 h.

Grade 8 use notes

• Unless a problem asks for an exact form, round decimal answers to the nearest tenth.
• Use pi ~= 3.14 or the calculator pi key consistently.
• Label units carefully: cm^2 for area, cm^3 for volume, and cm for height or radius.

Difficulty tags

• A = foundational
• B = developing
• C = proficient
• D = challenge

Practice Set

Set A: direct layering and volume fluency

1. A cylinder has radius 3 cm and height 8 cm. Find its volume.
Tags: A, direct computation, exact or decimal

2. A cylinder has radius 5 cm and height 4 cm. Find its volume.
Tags: A, direct computation

3. A cylinder has diameter 10 cm and height 7 cm. Find its volume.
Tags: A, diameter to radius

4. A cylinder has radius 2.5 m and height 6 m. Find its volume.
Tags: A, decimal measurement

5. One circular layer of a cylinder has area 49pi cm^2. The cylinder is 12 cm tall. Find the volume.
Tags: A, layer area to volume

6. A can is modelled by a cylinder with radius 4 cm and height 15 cm. Find its volume.
Tags: A, application

Set B: interpreting the layer model

7. Explain in one or two sentences why multiplying the area of the base by the height gives the volume of a cylinder.
Tags: B, visualization, explanation

8. A cylinder is made of 9 equal circular layers. Each layer has area 16pi cm^2. What is the volume?
Tags: B, discrete layers, structure

9. A cylinder has height 11 cm. Each horizontal slice has area 20pi cm^2. What is the volume?
Tags: B, layer interpretation

10. Two cylinders have the same height, 10 cm. Cylinder P has radius 2 cm. Cylinder Q has radius 4 cm. How many times as large is the volume of Q compared with P?
Tags: B, comparison, reasoning about r^2

11. Two cylinders have the same radius, 3 cm. Cylinder R has height 5 cm. Cylinder S has height 15 cm. Compare their volumes.
Tags: B, comparison, scaling height

12. A cylinder has radius 7 cm and height 2 cm. Another has radius 7 cm and height 9 cm. Find both volumes and the difference.
Tags: B, same base different height

Set C: reverse problems and mixed skills

13. A cylinder has volume 200pi cm^3 and height 8 cm. Find the radius.
Tags: C, reverse, solve for radius

14. A cylinder has volume 768pi m^3 and radius 8 m. Find the height.
Tags: C, reverse, solve for height

15. A cylinder has diameter 14 cm and volume 686pi cm^3. Find the height.
Tags: C, diameter, reverse

16. A cylinder has radius r and height h. If the radius doubles and the height stays the same, by what factor does the volume change?
Tags: C, generalization, algebraic reasoning

Set D: challenge and multi-step application

17. A soup container is a cylinder with diameter 8 cm and height 12 cm. How much soup can it hold? Give an exact answer in terms of pi and a decimal approximation.
Tags: D, application, diameter to radius

18. A cylinder has volume 490pi cm^3 and radius 7 cm. Find its height, then explain what that height means in the layer model.
Tags: D, reverse, concept explanation

19. Cylinder A has radius 3 cm and height 20 cm. Cylinder B has radius 6 cm and height 5 cm. Which cylinder has greater volume, or are they equal? Justify with calculations.
Tags: D, comparison, non-obvious

20. A cylindrical water tank has radius 1.5 m and height 3.2 m. Find its volume to the nearest tenth of a cubic metre.
Tags: D, real-world, decimal computation

21. A cylinder has the same volume as another cylinder, but its radius is half as large. How must its height change to keep the volume the same?
Tags: D, invariant volume, reasoning

22. A cylinder has base area 81pi cm^2 and volume 972pi cm^3. Find its radius and height.
Tags: D, multi-step reverse

23. A student says, "If the height doubles, the volume doubles, and if the radius doubles, the volume also doubles." Identify the correct part, identify the error, and repair the statement.
Tags: D, misconception repair, reasoning

24. A candle is shaped like a cylinder with radius 2.8 cm and height 15 cm. Wax fills the whole cylinder. If 1 cm^3 of wax has mass 0.9 g, find the mass of the candle to the nearest gram.
Tags: D, modelling, multi-step application

Answer Key

1. 72pi cm^3 (~226.2 cm^3)
2. 100pi cm^3 (~314.2 cm^3)
3. 175pi cm^3 (~549.8 cm^3)
4. 37.5pi m^3 (~117.8 m^3)
5. 588pi cm^3
6. 240pi cm^3 (~754.0 cm^3)
7. Because each horizontal layer is the same circle, the cylinder is a stack of equal circular layers; multiplying one layer's area by the total height gives the total space inside.
8. 144pi cm^3
9. 220pi cm^3
10. 4 times
11. Cylinder S has 3 times the volume of Cylinder R.
12. 98pi cm^3 and 441pi cm^3; difference 343pi cm^3
13. 5 cm
14. 12 m
15. 14 cm
16. The volume is multiplied by 4.
17. 384pi cm^3 (~1206.4 cm^3)
18. 10 cm; it is the number of 1 cm-thick layers, or the stack height of equal circular layers.
19. The volumes are equal: both are 180pi cm^3.
20. 22.6 m^3
21. The height must become 4 times as large.
22. Radius 9 cm, height 12 cm
23. Doubling height does double volume; doubling radius multiplies volume by 4, not 2.
24. About 333 g

Full Solutions for the Hardest Third

17. Soup container

Given diameter 8 cm, so radius r = 4 cm. Height h = 12 cm.

Use V = pi r^2 h.

V = pi(4^2)(12)
V = pi(16)(12)
V = 192pi

So the container holds 192pi cm^3.

Decimal approximation:
192pi ~= 603.2

Answer: 192pi cm^3 or about 603.2 cm^3.

18. Reverse volume and layer meaning

Given V = 490pi cm^3 and r = 7 cm.

Use V = pi r^2 h.

490pi = pi(7^2)h
490pi = 49pi h

Divide both sides by 49pi:

h = 10

So the height is 10 cm.

Layer meaning: each horizontal layer has area pi(7^2) = 49pi cm^2, and stacking those equal layers through 10 cm gives the total volume 490pi cm^3.

Answer: 10 cm; it represents the full stacked layer height.

19. Compare two cylinders

Cylinder A: r = 3 cm, h = 20 cm

V_A = pi(3^2)(20) = pi(9)(20) = 180pi cm^3

Cylinder B: r = 6 cm, h = 5 cm

V_B = pi(6^2)(5) = pi(36)(5) = 180pi cm^3

Since V_A = V_B, the cylinders have equal volume.

This is a useful comparison: Cylinder B has double the radius, which tends to increase volume a lot, but it also has only one quarter of the height compared with the effect on r^2, so the totals balance.

Answer: The cylinders are equal in volume.

20. Cylindrical water tank

Given r = 1.5 m, h = 3.2 m.

V = pi r^2 h
V = pi(1.5^2)(3.2)
V = pi(2.25)(3.2)
V = 7.2pi

Now approximate:

7.2pi ~= 22.619...

Rounded to the nearest tenth:

V ~= 22.6 m^3

Answer: 22.6 m^3

21. Same volume, half the radius

Let the original cylinder have volume

V = pi r^2 h

The new cylinder has radius r/2 and height H.

Its volume is

V = pi(r/2)^2 H = pi(r^2/4)H

To keep the volumes equal:

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

Divide both sides by pi r^2:

h = H/4

So

H = 4h

The height must be four times as large because halving the radius makes the base area one quarter as large.

Answer: The height must be multiplied by 4.

22. Find both radius and height

Base area is 81pi cm^2.

Since base area of a cylinder is pi r^2:

pi r^2 = 81pi

Divide by pi:

r^2 = 81

r = 9 cm

Now use the volume 972pi cm^3:

V = pi r^2 h
972pi = pi(9^2)h
972pi = 81pi h

Divide by 81pi:

h = 12 cm

Answer: Radius 9 cm, height 12 cm.

23. Repairing a misconception

The student's first claim is correct: if height doubles and radius stays the same, then

V = pi r^2 h

shows that volume doubles because only h changes by a factor of 2.

The second claim is incorrect. If radius doubles, then

V = pi(2r)^2 h = pi(4r^2)h = 4pi r^2 h

So the volume becomes 4 times as large, not 2 times as large.

A correct repaired statement is:

• Doubling the height doubles the volume.
• Doubling the radius quadruples the volume.

Answer: Height claim correct; radius claim false because the radius is squared.

24. Candle mass

Given r = 2.8 cm, h = 15 cm.

First find volume:

V = pi r^2 h
V = pi(2.8^2)(15)
V = pi(7.84)(15)
V = 117.6pi cm^3

Approximate:

117.6pi ~= 369.5 cm^3

Now use mass = 0.9 g per 1 cm^3:

mass ~= 369.5 x 0.9 = 332.55 g

Round to the nearest gram:

333 g

Answer: About 333 g.

Common mistakes to watch for

• Using diameter as radius. Repair: divide the diameter by 2 before squaring.
• Forgetting to square the radius. Repair: in pi r^2 h, only r is squared.
• Writing square units for volume. Repair: volume must use cubic units such as cm^3.
• Assuming all dimensions scale volume the same way. Repair: height changes volume linearly, but radius changes it quadratically.

Suggested checking routine

1. Identify whether the measurement given is a radius or a diameter.
2. Write V = pi r^2 h before substituting.
3. Estimate whether the answer should be small, medium, or large.
4. Check units at the end.
5. For reverse problems, substitute your answer back into the formula.