Volume Formula Fundamentals: Graded Practice Set

Position in the unit

This record is for Grade 8 Geometry and Measurement in the Canadian curriculum and is meant for deliberate practice after these linked records:

• rea.m08.geometry-measurement.volume.basic-formulas.overview
• rea.m08.geometry-measurement.volume.basic-formulas.lesson
• rea.m08.geometry-measurement.volume.basic-formulas.worked-examples

Use those records for the full teaching of why V = Bh, V = lwh, and V = pi r^2 h work. This record focuses on practice, checking, and reverse problems.

How to use this set

• Work in order; the difficulty rises gradually.
• Keep units visible at every step.
• For cylinders, leave answers in terms of pi unless the question asks for an approximation.
• For missing-dimension problems, solve algebraically and then check by substitution.

Difficulty tags

• D1 = direct substitution
• D2 = reverse or comparison problem
• D3 = unit conversion or multi-step setup
• D4 = challenge application

Practice problems

1. D1 A right rectangular prism has length 4 cm, width 3 cm, and height 5 cm. Find its volume.
2. D1 A box measures 9 m by 2 m by 6 m. Find its volume.
3. D1 A prism measures 12.5 cm by 4 cm by 2 cm. Find its volume.
4. D1 A right prism has base area 18 cm^2 and height 7 cm. Find its volume.
5. D1 A cylinder has radius 3 cm and height 5 cm. Find its volume in exact form and as an approximation using pi ≈ 3.14.
6. D1 A cylinder has diameter 10 cm and height 12 cm. Find its volume in exact form and as an approximation.
7. D2 A rectangular prism has volume 960 cm^3, length 15 cm, and width 8 cm. Find its height.
8. D2 A rectangular prism has volume 504 cm^3, width 7 cm, and height 9 cm. Find its length.
9. D2 A cylinder has volume 196pi cm^3 and height 4 cm. Find its radius.
10. D2 Which holds more volume: a cube with side length 6 cm, or a cylinder with radius 3 cm and height 8 cm?
11. D3 A rectangular container is 0.4 m long, 25 cm wide, and 30 cm high. Find its volume in cm^3.
12. D3 A storage bin is 0.8 m long, 0.25 m wide, and 0.4 m high. Find its volume in m^3.
13. D2 A right prism has base area 32.4 cm^2 and height 6.5 cm. Find its volume.
14. D2 A cylinder has radius 2.5 cm and height 10 cm. Find its volume in exact form and as an approximation.
15. D2 A rectangular prism has volume 157.5 cm^3, length 7.5 cm, and height 3 cm. Find its width.
16. D3 A cylinder has diameter 0.6 m and height 120 cm. Find its volume in m^3 in exact form and as an approximation.
17. D3 A rectangular prism has volume 472.5 cm^3, length 10.5 cm, and width 6 cm. Find its height.
18. D3 A cylinder has diameter 14 cm and volume 1176pi cm^3. Find its height.
19. D3 A right prism has base area 28 cm^2, height 9 cm, and another slanted edge marked 10 cm on the diagram. Find the volume, and identify which measurement was unnecessary.
20. D3 Which container holds more: a rectangular prism 30 cm x 20 cm x 15 cm, or a cylinder with radius 8 cm and height 45 cm? How much more does it hold?
21. D4 A cylinder has volume 0.4pi m^3 and height 2.5 m. Find its diameter.
22. D4 A cylindrical tank has radius 0.5 m and height 1.2 m. It already contains 0.18 m^3 of water. How much more water is needed to fill it completely? Give an exact expression and an approximation.
23. D4 A shipping crate is a rectangular prism 50 cm x 20 cm x 10 cm. Three identical smaller boxes, each 20 cm x 15 cm x 8 cm, are placed inside. Assuming volume is the only consideration, how much space remains unused?
24. D4 A block of wax is a rectangular prism 18 cm x 10 cm x 7 cm. It is melted and poured into a cylinder with radius 3 cm. What height will the cylinder have? Give an exact answer in terms of pi and an approximation.

Answer key

1. 60 cm^3
2. 108 m^3
3. 100 cm^3
4. 126 cm^3
5. 45pi cm^3 ≈ 141.3 cm^3
6. 300pi cm^3 ≈ 942 cm^3
7. 8 cm
8. 8 cm
9. 7 cm
10. The cylinder; 72pi cm^3 ≈ 226.08 cm^3 versus 216 cm^3, so it holds about 10.08 cm^3 more.
11. 30000 cm^3
12. 0.08 m^3
13. 210.6 cm^3
14. 62.5pi cm^3 ≈ 196.25 cm^3
15. 7 cm
16. 0.108pi m^3 ≈ 0.33912 m^3
17. 7.5 cm
18. 24 cm
19. 252 cm^3; the 10 cm slanted edge was unnecessary.
20. The cylinder; 2880pi cm^3 ≈ 9043.2 cm^3 versus 9000 cm^3, so it holds about 43.2 cm^3 more.
21. 0.8 m
22. 0.3pi - 0.18 m^3 ≈ 0.762 m^3
23. 2800 cm^3
24. 140/pi cm ≈ 44.6 cm

Full solutions for the hardest third

17. Rectangular prism with a missing height

Given:

• V = 472.5 cm^3
• l = 10.5 cm
• w = 6 cm

Use V = lwh.

472.5 = 10.5 x 6 x h

First multiply the known dimensions:

10.5 x 6 = 63

So:

472.5 = 63h

Now divide by 63:

h = 472.5 / 63 = 7.5

Answer: 7.5 cm

Check:

10.5 x 6 x 7.5 = 63 x 7.5 = 472.5

The check works.

18. Cylinder with a missing height

Given:

• diameter = 14 cm
• volume = 1176pi cm^3

The radius is half the diameter:

r = 7 cm

Use V = pi r^2 h.

1176pi = pi x 7^2 x h

1176pi = 49pi h

Divide both sides by 49pi:

h = 1176pi / 49pi = 1176 / 49 = 24

Answer: 24 cm

Check:

pi x 7^2 x 24 = pi x 49 x 24 = 1176pi

19. Right prism with extra information

Given:

• base area B = 28 cm^2
• height h = 9 cm
• another slanted edge = 10 cm

Use V = Bh because the base area is already known.

V = 28 x 9 = 252

Answer: 252 cm^3

The unnecessary measurement was the 10 cm slanted edge. Volume only needs base area and perpendicular height here.

20. Comparing two containers

Rectangular prism:

V = lwh = 30 x 20 x 15 = 9000 cm^3

Cylinder:

V = pi r^2 h = pi x 8^2 x 45

V = pi x 64 x 45 = 2880pi cm^3

Approximate:

2880pi ≈ 2880 x 3.14 = 9043.2 cm^3

Compare:

9043.2 - 9000 = 43.2

Answer: The cylinder holds more, by about 43.2 cm^3.

21. Cylinder with volume and height given

Given:

• V = 0.4pi m^3
• h = 2.5 m

Use V = pi r^2 h.

0.4pi = pi r^2 x 2.5

Divide both sides by pi:

0.4 = 2.5r^2

Now divide by 2.5:

r^2 = 0.4 / 2.5 = 0.16

So:

r = 0.4 m

Diameter is twice the radius:

d = 0.8 m

Answer: 0.8 m

22. Tank not yet full

Total tank volume:

V = pi r^2 h = pi x 0.5^2 x 1.2

0.5^2 = 0.25

So:

V = pi x 0.25 x 1.2 = 0.3pi m^3

Water already in tank: 0.18 m^3

Water still needed:

0.3pi - 0.18 m^3

Approximate:

0.3pi ≈ 0.3 x 3.14 = 0.942

0.942 - 0.18 = 0.762

Answer: 0.3pi - 0.18 m^3 ≈ 0.762 m^3

23. Unused space in a crate

First find the crate volume:

V = 50 x 20 x 10 = 10000 cm^3

Now find one small box volume:

V = 20 x 15 x 8 = 2400 cm^3

Three boxes use:

3 x 2400 = 7200 cm^3

Unused space:

10000 - 7200 = 2800 cm^3

Answer: 2800 cm^3

24. Converting prism volume into cylinder height

The wax volume stays the same.

Rectangular prism volume:

V = 18 x 10 x 7 = 1260 cm^3

For the cylinder, use V = pi r^2 h with r = 3 cm.

1260 = pi x 3^2 x h

1260 = 9pi h

Solve for h:

h = 1260 / 9pi = 140/pi

Approximate:

h ≈ 140 / 3.14 ≈ 44.6

Answer: 140/pi cm ≈ 44.6 cm

Common mistakes to watch for while checking

• Using diameter in place of radius in the cylinder formula.
• Writing square units instead of cubic units.
• Forgetting to convert all measurements to the same unit before substituting.
• Using an extra number from the diagram that does not describe base area or perpendicular height.
• Rounding too early when pi is involved.

Suggested next use

After this set, a learner should be ready for mixed review that combines volume with surface area, or for more advanced records on composite solids and reverse measurement problems.