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: A cone is one-third of a cylinder—tapering to a point reduces the volume by a factor of 3.

Common stuck point: The \frac{1}{3} factor applies to all pyramids and cones, not just circular ones.

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 \pi.

Solution

  1. 1
    Step 1: Write the formula: V = \frac{1}{3}\pi r^2 h.
  2. 2
    Step 2: Substitute: V = \frac{1}{3}\pi (6)^2(9) = \frac{1}{3}\pi \times 36 \times 9 = \frac{1}{3} \times 324\pi.
  3. 3
    Step 3: Simplify: V = 108\pi cm³.

Answer

V = 108\pi cm³.
The cone formula V = \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 \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 75\pi cm³ and a radius of 5 cm. Find the height of the cone.

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 90\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.)
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Background Knowledge

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

volume of cylinder