Volume of Cones
A cone with the same base and height as a cylinder holds exactly one third as much. That one fact - plus spotting the true height - solves almost every cone problem.
Know these first
- Volume of Cylinders 8.G.9
This unlocks
- Volume of Spheres 8.G.9
What you'll be able to do
- Find the volume of a cone with V = ⅓πr²h, in terms of π or rounded.
- Work backward to a missing height or radius.
- Find a cone's height from its slant height with the Pythagorean theorem.
- Find the volume of shapes built from cones, cylinders and hemispheres.
The big idea
Fill a cone with sand and pour it into a cylinder with the same base and height. It takes exactly three cones to fill the cylinder.
So a cone's volume is one third of the cylinder's: V = ⅓πr²h.
Careful with the height: h is the straight-up height from the tip to the center of the base, not the slanted side. If you're given the slant height, the height, radius and slant make a right triangle - use the Pythagorean theorem first.
Words you need
- Apex
- The pointed tip of the cone.
- Height (h)
- The straight-up distance from the apex to the center of the base.
- Slant height
- The distance from the apex down the side to the edge of the base.
- Composite solid
- A shape made of simpler solids joined together, or with a piece cut out.
Worked examples
Example 1
A cone has radius 3 and height 4. Find its volume in terms of π.
- The base area is π × 3² = 9π.
- A cylinder would hold 9π × 4 = 36π.
- A cone holds one third of that: 36π ÷ 3 = 12π.
Answer: 12π cubic units
Example 2
A cone has radius 5 and slant height 13. Find its volume in terms of π.
- The height, radius and slant height form a right triangle: h² + 5² = 13².
- h² = 169 − 25 = 144, so h = 12.
- V = ⅓ × π × 5² × 12 = ⅓ × 300π = 100π.
Answer: 100π cubic units
Example 3
A cone has radius 6 and volume 96π. Find its height.
- Start from V = ⅓πr²h: 96π = ⅓ × π × 36 × h.
- So 96π = 12πh.
- Divide by 12π: h = 8.
Answer: h = 8 units
Common mistakes
Forgetting the ⅓. A cone holds one third of the matching cylinder. Leave out the ⅓ and your answer is 3 times too big.
Using the slant height as the height. h goes straight up through the middle. If you're given the slant, find h with the Pythagorean theorem first.
Using the diameter as the radius. Halve the distance across before you square it.
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
- Find the volume of the cone in terms of π.
- Find the volume of the cone in terms of π.
- An ice cream cone has radius 2 inches and height 6 inches. What is its volume in terms of π? Word problem
- A funnel is a cone with radius 4 cm and height 9 cm. What is its volume, to the nearest tenth of a cubic centimeter? Word problem
- A party hat is a cone with radius 3 inches and height 8 inches. What is its volume in terms of π? Word problem
Tier 2 Intermediate One more step to think through
- A cone has radius 3 and volume 30π. Find its height.
- A cone has height 6 and volume 50π. Find its radius.
- Find the volume of the cone in terms of π.
- A pile of sand forms a cone with radius 5 feet and height 3 feet. How much sand is there, to the nearest tenth of a cubic foot? Word problem
- A traffic cone has radius 3.5 inches and height 18 inches. What is its volume in terms of π? Word problem
- A paper cup shaped like a cone has radius 2 inches and holds 8π cubic inches. How tall is it? Word problem
- A cone-shaped tent has radius 9 feet and slant height 15 feet. What is its volume in terms of π? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A cone's radius is doubled and its height stays the same. How many times as large is the new volume?
- An ice cream cone has radius 3 cm and height 10 cm, with a perfect half-sphere scoop of radius 3 cm on top. What is the total volume in terms of π? Word problem
- A cylinder and a cone both have radius 4 cm and height 9 cm. How many cones of water would it take to fill the cylinder? Word problem
- A cone-shaped pile of salt is 6 meters tall and holds 128π cubic meters. What is its radius? Word problem
- A funnel is a cone with radius 3 cm and height 4 cm, attached to a spout that is a cylinder with radius 1 cm and length 5 cm. What is the total volume in terms of π? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- A wooden cylinder has radius 3 inches and height 9 inches. A cone with the same base and height is carved out of it. How much wood is left, in terms of π? Word problem
- A cone has a slant height of 17 cm and a radius of 8 cm. What is its volume in terms of π? Word problem
- A cone-shaped tank with radius 6 feet and height 10 feet is filled at 5 cubic feet per minute. How many minutes does it take to fill, to the nearest minute? Word problem
Answers
Tier 1 Foundations
- 21π cubic units. V = ⅓πr²h = 21π.
- 24π cubic units. V = ⅓πr²h = 24π.
- 8π cubic inches. V = ⅓πr²h = 8π.
- about 150.8 cubic cm. V = ⅓πr²h = 48π ≈ 150.8.
- 24π cubic inches. V = ⅓πr²h = 24π.
Tier 2 Intermediate
- h = 10 units. Start from V = ⅓πr²h, put in what you know, and solve for h.
- r = 5 units. Start from V = ⅓πr²h, put in what you know, and solve for r.
- 96π cubic units. h = √(10² − 6²) = 8, then V = ⅓πr²h = 96π.
- about 78.5 cubic feet. V = ⅓πr²h = 25π ≈ 78.5.
- 73.5π cubic inches. V = ⅓πr²h = 73.5π.
- h = 6 inches. Start from V = ⅓πr²h, put in what you know, and solve for h.
- 324π cubic feet. h = √(15² − 9²) = 12, then V = ⅓πr²h = 324π.
Tier 3 Advanced
- 4 times as much. In V = ⅓πr²h, r is squared and h is not: 2² × 1 = 4.
- 48π cubic cm. Add the pieces: cone 30π + hemisphere 18π = 48π.
- 3. The cylinder holds 144π and the cone holds 48π, so it takes 144π ÷ 48π = 3 cones.
- r = 8 meters. Start from V = ⅓πr²h, put in what you know, and solve for r.
- 17π cubic cm. Add the pieces: cone 12π + cylinder 5π = 17π.
Tier 4 Challenge
- 54π cubic inches. Add the pieces: cylinder 81π − cone 27π = 54π.
- 320π cubic cm. h = √(17² − 8²) = 15, then V = ⅓πr²h = 320π.
- about 75 minutes. V = ⅓ × π × 36 × 10 = 120π ≈ 377.0 cubic feet. At 5 cubic feet per minute that's about 75 minutes.
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