Grade 8 Link to Prism Volume: Graded Practice Set

Position in the Rea sequence

Use this set after:

• rea.m08.geometry-measurement.volume.cylinder-volume-derivation-practice-set for the full derivation of cylinder volume
• rea.m08.geometry-measurement.volume.cylinder-volume-formulas.practice-set for direct formula fluency
• rea.m08.geometry-measurement.volume.basic-formulas.practice-set for mixed base-area-times-height work
• rea.m08.geometry-measurement.volume.basic-formulas.formula-selection.practice-set for choosing the correct volume model

This record does not restate those lessons. Here, the goal is to practice the specific bridge:

prism volume = base area x height and cylinder volume = base area x height

The only structural difference is the shape of the base.

How to use this set

For each problem:

1. Identify the base.
2. Find its area.
3. Multiply by the perpendicular height.
4. If asked, explain the connection to prism volume in words.

Give exact answers in terms of pi unless a decimal approximation is requested.

Difficulty tags

• E = easy
• M = medium
• H = hard

Problems

1. E A right rectangular prism has base area 18 cm^2 and height 7 cm. Find its volume.
2. E A cylinder has circular base area 18 cm^2 and height 7 cm. Find its volume.
3. E Explain in one sentence why your calculations in Problems 1 and 2 use the same structure.
4. E A prism and a cylinder both have height 10 m. Their bases each have area 12 m^2. Find each volume.
5. E A cylinder has radius 3 cm and height 5 cm. Find its volume in exact form.
6. E Show how Problem 5 fits the rule base area x height by first finding the circular base area.
7. E A prism has volume 96 in^3 and base area 12 in^2. Find its height.
8. E A cylinder has volume 96pi cm^3 and base area 16pi cm^2. Find its height.
9. M A cylinder and a prism have the same height, 8 cm. The prism base is a rectangle 4 cm x 5 cm. The cylinder base has radius 2 cm. Which solid has greater volume?
10. M A cylinder has diameter 10 cm and height 9 cm. Find its volume in exact form.
11. M A circular base has area 49pi cm^2. A cylinder built on that base has height 6 cm. Find the cylinder's volume.
12. M A prism and a cylinder have equal volumes and equal heights of 11 cm. The prism base area is 30 cm^2. What is the cylinder's base area?
13. M A cylinder has the same base area and height as a prism whose volume is 180 cm^3. Find the cylinder's volume and justify your answer.
14. M A cylinder has radius 4 m. A prism has base area equal to the area of the cylinder's base. If both heights are 12 m, find the volume of each solid.
15. M A cylinder has volume 200pi cm^3 and height 8 cm. Find the area of its base. Then find the radius.
16. M A prism has base area 24 cm^2 and height h. A cylinder has base area 6pi cm^2 and the same height h. If their volumes are equal, find h or explain why it cannot be determined.
17. H A cylinder and a right rectangular prism have the same volume and the same height of 15 cm. The prism's base is 6 cm x 10 cm. Find the radius of the cylinder.
18. H A cylinder is filled with water. Its volume equals the volume of a prism with base area 45 cm^2 and height 8 cm. The cylinder's height is also 8 cm. Find the cylinder's radius in exact form.
19. H A prism has base area B. A cylinder has radius r and the same height h. If the two solids have equal volume, write an equation relating B and r, then solve for B.
20. H A cylinder and a prism each have volume 154pi cm^3. The cylinder has radius 3.5 cm. Find the cylinder's height. Then state one condition the prism must satisfy in order to have the same volume.
21. H A cylinder has radius 6 cm. A prism has rectangular base 9 cm x 4 cm. The two solids have equal volume. Find the ratio of the prism's height to the cylinder's height in simplest form.
22. H A company makes two containers with equal height 20 cm: one is a cylinder of radius 5 cm, and one is a prism with square base of side 9 cm. Which container holds more, and by how much? Give an exact answer and a decimal approximation to the nearest tenth cubic centimetre.
23. H A student says, "A cylinder uses a different volume idea than a prism because one has a curved side." Critique the statement using base-area-times-height reasoning and one numerical example of your choice.
24. H A cylinder is redesigned so that its height is cut in half while its base area is doubled. Compare the new volume to the original volume. Then explain how the same reasoning works for a prism.

Answer key

1. 126 cm^3
2. 126 cm^3
3. Both use volume = base area x height; only the base shape changes.
4. Each volume is 120 m^3.
5. 45pi cm^3
6. Base area = pi(3)^2 = 9pi cm^2, so volume = 9pi x 5 = 45pi cm^3.
7. 8 in
8. 6 cm
9. Prism: 160 cm^3; cylinder: 32pi cm^3, so the prism is greater.
10. 225pi cm^3
11. 294pi cm^3
12. 30 cm^2
13. 180 cm^3; same base area and same height imply same volume.
14. Each volume is 192pi m^3.
15. Base area 25pi cm^2; radius 5 cm.
16. h cannot be determined; equal volumes only imply 24h = 6pih, which compares base areas, not a unique height.
17. Radius sqrt(60/pi) cm
18. Radius 3sqrt(5/pi) cm
19. Bh = pi r^2 h, so B = pi r^2 (for h ≠ 0).
20. Height 4 cm; the prism must satisfy base area x height = 154pi cm^3.
21. prism height : cylinder height = pi : 1
22. The prism holds more by 120(81 - 25pi) cm^3, about 292.5 cm^3.
23. The statement is false; both use the same structure base area x height, and a valid numerical example will show matching reasoning.
24. The volume stays the same; doubling base area and halving height keeps the product unchanged for both cylinders and prisms.

Full solutions for the hardest third

17. Same volume, same height

The prism's base area is:

6 x 10 = 60 cm^2

So the prism's volume is:

60 x 15 = 900 cm^3

The cylinder has the same volume and the same height 15 cm, so:

pi r^2 (15) = 900

Divide by 15:

pi r^2 = 60

Now solve for r:

r^2 = 60/pi

r = sqrt(60/pi)

So the cylinder's radius is:

sqrt(60/pi) cm

18. Matching a prism by base-area-times-height

The prism's volume is:

45 x 8 = 360 cm^3

The cylinder has the same volume and the same height 8 cm, so:

pi r^2 (8) = 360

Divide by 8:

pi r^2 = 45

Solve for r^2:

r^2 = 45/pi

Take the square root:

r = sqrt(45/pi) = 3sqrt(5/pi)

So the radius is:

3sqrt(5/pi) cm

19. General algebra link between prism and cylinder

Prism volume:

V = Bh

Cylinder volume:

V = pi r^2 h

If the volumes are equal, then:

Bh = pi r^2 h

If h is nonzero, divide both sides by h:

B = pi r^2

This shows the link directly: for equal heights and equal volumes, the prism's base area must equal the cylinder's circular base area.

20. Reverse problem with a cylinder

For the cylinder:

V = pi r^2 h

Substitute V = 154pi and r = 3.5:

154pi = pi(3.5)^2 h

Compute (3.5)^2:

154pi = 12.25pi h

Divide by 12.25pi:

h = 154 / 12.25 = 4/0.318181... = 4

So the cylinder's height is:

4 cm

For a prism to have the same volume, it must satisfy:

base area x height = 154pi cm^3

There are many possible prisms; that product is the required condition.

21. Comparing heights from equal volumes

Cylinder volume:

V = pi(6)^2 h_c = 36pi h_c

Prism base area:

9 x 4 = 36 cm^2

Prism volume:

V = 36 h_p

Equal volumes mean:

36 h_p = 36pi h_c

Divide by 36:

h_p = pi h_c

So the ratio is:

h_p : h_c = pi : 1

This is a clean example of the prism-cylinder link: once base areas are known, height scales to preserve equal volume.

22. Comparing two containers

Cylinder volume:

V_c = pi r^2 h = pi(5)^2(20) = 500pi cm^3

Prism volume:

The square base area is:

9 x 9 = 81 cm^2

So:

V_p = 81 x 20 = 1620 cm^3

Compare them:

1620 - 500pi cm^3

Factor out 20 if desired:

20(81 - 25pi) cm^3

Since 1620 > 500pi, the prism holds more.

Using pi ≈ 3.1416:

500pi ≈ 1570.8

1620 - 1570.8 ≈ 49.2

Wait carefully: using the factored exact form,

20(81 - 25pi) ≈ 20(81 - 78.54) ≈ 20(2.46) ≈ 49.2

So the prism holds more by:

1620 - 500pi cm^3 ≈ 49.2 cm^3

23. Critiquing the misconception

The statement is false. A curved side does not change the volume structure. Volume still measures how much space is inside the solid, and for both prisms and cylinders the amount of space is found by multiplying:

base area x height

The side shape changes the appearance, but not that structure.

Example:

• Prism with base area 12 cm^2 and height 7 cm has volume 12 x 7 = 84 cm^3.
• Cylinder with base area 12 cm^2 and height 7 cm also has volume 12 x 7 = 84 cm^3.

So the cylinder does not need a different volume idea; it uses the same one with a circular base.

24. Changing base area and height

Original volume:

V = Bh

For the cylinder, B stands for the circular base area. After the redesign:

• new base area = 2B
• new height = h/2

New volume:

V_new = (2B)(h/2)

V_new = Bh

So the new volume is the same as the original volume.

Exactly the same reasoning works for a prism because prism volume is also base area x height. The base shape does not matter here; only the product matters.

Common error checks

• Do not multiply pi r^2 by itself; that is base area squared, not volume.
• Do not use the diameter as the radius.
• If two solids have the same base area and the same height, then they have the same volume, even if one has curved sides.
• If heights are equal and volumes are equal, then base areas must be equal.
• Keep exact answers in terms of pi unless the problem asks for a decimal.