Volume of Spheres
A sphere's volume depends only on its radius - and because the radius is cubed, a ball twice as wide holds 8 times as much.
Know these first
- Volume of Cylinders 8.G.9
- Volume of Cones 8.G.9
This unlocks
- More lessons that build on this are coming.
What you'll be able to do
- Find the volume of a sphere with V = ⁴⁄₃πr³, in terms of π or rounded.
- Find the volume of a hemisphere (half a sphere).
- Work backward from a volume to the radius.
- Compare spheres, cylinders and cones, and handle shapes built from them.
The big idea
Put a ball snugly inside a can: the can's radius matches the ball's, and its height is the ball's diameter, 2r.
The can holds πr² × 2r = 2πr³. The ball fills exactly two thirds of it: ⅔ × 2πr³ = ⁴⁄₃πr³. That's the sphere's volume.
Because r is cubed, size matters fast. Double the radius and the volume becomes 2³ = 8 times as big.
Words you need
- Radius (r)
- The distance from the center to the surface.
- Diameter
- All the way through the center: twice the radius.
- Hemisphere
- Half a sphere. Its volume is ⅔πr³.
- Cubed
- Multiplied by itself three times: r³ = r × r × r.
Worked examples
Example 1
A sphere has radius 3. Find its volume in terms of π.
- Cube the radius: 3³ = 27.
- Multiply by ⁴⁄₃: ⁴⁄₃ × 27 = 36.
- V = 36π.
Answer: 36π cubic units
Example 2
A ball is 10 cm across. Find its volume to the nearest tenth.
- 10 cm across is the diameter, so r = 5 cm.
- V = ⁴⁄₃ × π × 5³ = ⁴⁄₃ × 125π ≈ 166.67π.
- 166.67 × 3.14159 ≈ 523.6 cubic centimeters.
Answer: about 523.6 cubic cm
Example 3
A sphere has volume 288π. Find its radius.
- Start from ⁴⁄₃πr³ = 288π.
- Divide by π and multiply by ¾: r³ = 216.
- r = 6, because 6 × 6 × 6 = 216.
Answer: r = 6 units
Common mistakes
Using the diameter as the radius. Halve the distance across first. A 10 cm ball has r = 5 cm.
Squaring instead of cubing. It's r³, not r². Volume fills three directions.
Forgetting the ⁴⁄₃ - or using ⅓. ⅓ is for cones. Spheres use ⁴⁄₃.
Practice: 20 questions
14 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- A sphere has radius 6. Find its volume in terms of π.
- A sphere has radius 1. Find its volume to the nearest tenth.
- A beach ball has a radius of 9 inches. What is its volume in terms of π? Word problem
- A marble has a radius of 0.5 cm. What is its volume, to the nearest tenth of a cubic centimeter? Word problem
- A basketball is 24 cm across. What is its volume, to the nearest tenth of a cubic centimeter? Word problem
Tier 2 Intermediate One more step to think through
- A sphere has volume 36π. Find its radius.
- A hemisphere has radius 3. Find its volume in terms of π.
- A sphere has a diameter of 6. Find its volume in terms of π.
- A gumball has a radius of 1.5 cm. What is its volume in terms of π? Word problem
- A dome is a hemisphere with radius 30 feet. How much space is under it, to the nearest tenth of a cubic foot? Word problem
- A balloon holds 288π cubic centimeters of air. What is its radius? Word problem
- An orange is 8 cm across. What is its volume, to the nearest tenth of a cubic centimeter? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A sphere's radius is doubled. How many times as large is its volume?
- A can is a cylinder with radius 3 cm and height 18 cm. It holds 3 tennis balls, each with radius 3 cm. How much empty space is left in the can, in terms of π? Word problem
- A grain silo is a cylinder with radius 4 m and height 20 m, topped by a hemisphere dome with radius 4 m. What is its total volume, to the nearest tenth? Word problem
- A ball holds 4500π cubic inches. What is its radius? Word problem
- How many small marbles with radius 1 cm can be melted down to make one ball with radius 4 cm? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- A ball with radius 5 cm fits snugly inside a can with radius 5 cm and height 10 cm. What fraction of the can does the ball fill? Give a decimal to two places. Word problem
- An ice cream treat is a cone with radius 3 cm and height 8 cm, topped with a half-sphere scoop of radius 3 cm. What is the total volume in terms of π? Word problem
- A sphere's radius is cut in half. What fraction of the original volume does the new sphere have? Word problem
Answers
Tier 1 Foundations
- 288π cubic units. V = ⁴⁄₃πr³ = 288π.
- about 4.2 cubic units. V = ⁴⁄₃πr³ = 1.33333π ≈ 4.2.
- 972π cubic inches. V = ⁴⁄₃πr³ = 972π.
- about 0.5 cubic cm. V = ⁴⁄₃πr³ = 0.166667π ≈ 0.5.
- about 7238.2 cubic cm. V = ⁴⁄₃πr³ = 2304π ≈ 7238.2.
Tier 2 Intermediate
- r = 3 units. Start from V = ⁴⁄₃πr³, put in what you know, and solve for r.
- 18π cubic units. V = ⅔πr³ (half a sphere) = 18π.
- 36π cubic units. V = ⁴⁄₃πr³ = 36π.
- 4.5π cubic cm. V = ⁴⁄₃πr³ = 4.5π.
- about 56548.7 cubic feet. V = ⅔πr³ (half a sphere) = 18000π ≈ 56548.7.
- r = 6 cm. Start from V = ⁴⁄₃πr³, put in what you know, and solve for r.
- about 268.1 cubic cm. V = ⁴⁄₃πr³ = 85.3333π ≈ 268.1.
Tier 3 Advanced
- 8 times as much. In V = ⁴⁄₃πr³, r is cubed: 2³ = 8.
- 54π cubic cm. Add the pieces: cylinder 162π − sphere 36π − sphere 36π − sphere 36π = 54π.
- about 1139.4 cubic m. Add the pieces: cylinder 320π + hemisphere 42.6667π = 362.667π ≈ 1139.4.
- r = 15 inches. Start from V = ⁴⁄₃πr³, put in what you know, and solve for r.
- 64. Volume grows with r³, so the big ball holds (4 ÷ 1)³ = 64 times as much as one marble.
Tier 4 Challenge
- about 0.67. The ball is ⁴⁄₃ × 125π ≈ 166.67π and the can is 250π, so the ball fills 166.67 ÷ 250 = ⅔ ≈ 0.67 of it.
- 42π cubic cm. Add the pieces: cone 24π + hemisphere 18π = 42π.
- 0.125 times as much. In V = ⁴⁄₃πr³, r is cubed: 0.5³ = 0.125.
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