Volume of a Cone Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Volume of a Cone.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The amount of three-dimensional space inside a cone, which is exactly one-third the volume of a cylinder with the same base and height.

Imagine filling a cone-shaped paper cup with water and pouring it into a cylinder of the same width and height. You'd need to fill the cone exactly three times to fill the cylinder. A cone is a cylinder that 'tapers to a point,' losing two-thirds of its volume in the process.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Cone volume starts with cylinder volume and adjusts for tapering to a point.

Common stuck point: The procedure for volume of a cone is the easy part; the trap is forgetting the 1/31/3 factor. Asking "Does the solid taper to one point instead of having two equal circular bases?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the solid taper to one point instead of having two equal circular bases?

Worked Examples

Example 1

easy
A cone has a radius of 6 cm and a height of 9 cm. Find its volume. Leave your answer in terms of π\pi.

Answer

V=108πV = 108\pi cm³.

First step

1
Step 1: Write the formula: V=13πr2hV = \frac{1}{3}\pi r^2 h.

Full solution

  1. 2
    Step 2: Substitute: V=13π(6)2(9)=13π×36×9=13×324πV = \frac{1}{3}\pi (6)^2(9) = \frac{1}{3}\pi \times 36 \times 9 = \frac{1}{3} \times 324\pi.
  2. 3
    Step 3: Simplify: V=108πV = 108\pi cm³.
The cone formula V=13πr2hV = \frac{1}{3}\pi r^2 h is exactly one-third the volume of a cylinder with the same base and height. This factor of 13\frac{1}{3} can be demonstrated by filling a cone with water and pouring it into a cylinder three times to fill it completely.

Example 2

medium
An ice cream cone has a volume of 75π75\pi cm³ and a radius of 5 cm. Find the height of the cone.

Example 3

medium
An ice-cream cone has a circular opening with radius 44 cm and a depth of 1212 cm. How many cubic cm of ice cream fits inside (just the cone)?

Example 4

hard
A cone and a hemisphere both have radius rr. Their volumes are equal. Find the cone's height in terms of rr.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A cone and a cylinder have the same radius and height. The cylinder's volume is 90π90\pi cm³. What is the cone's volume?

Example 2

hard
A cone has a slant height of 13 cm and a radius of 5 cm. Find its volume. (Hint: use the Pythagorean theorem to find the height first.)

Example 3

easy
A cone has radius 22 and height 99. Find its volume in terms of π\pi.

Example 4

easy
A cone has radius 44 and height 66. Find its volume in terms of π\pi.

Example 5

easy
A cone has diameter 1010 and height 66. Find its volume in terms of π\pi.

Example 6

easy
A cone has radius 33 and height 44. Find its volume in terms of π\pi.

Example 7

easy
A cone has volume 24π24\pi and radius 66. Find its height.

Example 8

easy
A cone of radius 33 and height 55 has volume approximately what? Use π≈3.14\pi \approx 3.14.

Example 9

medium
A cone has volume 75π75\pi and height 99. Find its radius.

Example 10

medium
Two cones have radii 33 and 66 and the same height. Find the ratio of their volumes (small : large).

Example 11

medium
Two cones have the same radius but heights 44 and 1010. Find the ratio of their volumes.

Example 12

medium
A cone has slant height 1717 and radius 88. Find its volume in terms of π\pi.

Example 13

medium
A cone of radius 66 and height 1212 has its top third (by height) cut off by a plane parallel to the base. Find the volume of the small cone that was removed.

Example 14

medium
A cone has V=100πV = 100\pi and radius 55. Find its height.

Example 15

medium
A cone has base area 36π36\pi and height 55. Find its volume.

Example 16

medium
A cone is half-filled with water (by height). What fraction of the cone's volume is the water?

Example 17

medium
A cone has a base radius 66 and height 99. A cylinder is inscribed with the same height and a radius such that the cylinder's top fits exactly inside the cone at height hh... actually find the ratio of cone volume to a cylinder of the same base and height.

Example 18

hard
Water pours into a conical cup (point down) of radius 55 and height 1010. When the water depth is 44, what is the water volume in terms of π\pi?

Example 19

hard
A frustum has top radius 33, bottom radius 66, and height 88. Find its volume in terms of π\pi.

Example 20

hard
A solid cone of radius 66 and height 1212 is melted and recast as a cube. Find the side length of the cube (in terms of π\pi).

Example 21

hard
A cone has height equal to its diameter, with radius rr. Express its volume.

Example 22

hard
Water drains from a conical tank (point down) of radius 44 and height 1212. At what depth dd is the tank exactly half full by volume?

Example 23

challenge
Among all cones inscribed in a sphere of radius RR (with vertex at the top of the sphere and base parallel to a horizontal plane through the sphere), find the height of the cone with maximum volume.

Example 24

challenge
A sand timer is two cones joined point-to-point inside a cylinder of radius rr and total height 2h2h (each cone has height hh). What fraction of the cylinder's volume is sand-fillable (i.e., total cone volume)?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

volume of cylinder