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# Volume of a Cone
- URL: https://www.esheets.io/volume-of-a-cone/
- Published: 2026-08-16T14:35:49.000Z
- Updated: 2026-08-16T14:35:49.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

This worksheet practises using the formula for the volume of a cone.

You will need to calculate the volume (**1⁄3πr²h**) using the radius and perpendicular height. You will also practise using Pythagoras to find the perpendicular height when the slant height is given, and working backwards from a given volume to find missing lengths. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This worksheet generates questions on finding the volume of a cone.

The progression moves from straightforward formula substitution towards using Pythagoras, and finishes with finding missing lengths from a known volume.

#### What you’ll practise

- Calculating the volume of a cone.
- Understanding the difference between radius and diameter.
- Using Pythagoras to find the perpendicular height when the slant height is given.
- Working backwards to find the perpendicular height or radius when the volume is known.
- Giving exact answers in terms of π.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the formula and worked examples.

## Volume of a cone

Calculate the volume of the cones, or work backwards to find a missing length.

## Topic guide

### The formula

To find the volume of a cone, you need to know the radius of the base (**r**) and the perpendicular height (**h**).

**Volume = 1⁄3 πr²h**

### Radius, perpendicular height and slant height

It is very important to use the correct measurements:

- **Radius (r):** The distance from the centre of the circular base to the edge. If you are given the diameter, halve it.
- **Perpendicular height (h):** The straight vertical distance from the point (apex) to the centre of the base. This is the length used in the volume formula.
- **Slant height (l):** The distance along the sloping side of the cone. **Do not use this directly in the volume formula.**

### Worked example 1: Finding volume

*Find the volume of a cone with radius 4 cm and perpendicular height 9 cm. Give your answer to 1 decimal place.*

- Identify the measurements: **r = 4**, **h = 9**
- Use the formula: **V = 1⁄3 × π × 4² × 9**
- Calculate: **V = 1⁄3 × π × 16 × 9 = 48π**
- Calculate the decimal: 48 × π ≈ 150.796...
- Round to 1 decimal place: **150.8 cm³**

### Worked example 2: Using Pythagoras

If you are given the slant height (l) instead of the perpendicular height (h), you must use Pythagoras' theorem first.

*Find the volume of a cone with radius 5 cm and slant height 13 cm.*

- Use Pythagoras to find h: **h² = 13² - 5²**
- **h² = 169 - 25 = 144**
- **h = √144 = 12 cm**
- Now use the volume formula: V = 1⁄3 × π × 5² × 12 = 1⁄3 × π × 25 × 12 = **100π cm³**

### Exact answers in terms of π

Sometimes you will be asked to give your answer "in terms of π". This means you do not multiply by 3.142\. Instead, you leave π in your final answer, exactly as shown in the examples above (e.g. 48π or 100π).

### Reverse problems (working backwards)

You may be given the numerical volume and asked to find the perpendicular height or radius.

*Example: The volume of a cone is 300 cm³. Its radius is 5 cm. Find the perpendicular height.*

- Write the equation: **1⁄3π × 5² × h = 300**
- Simplify: **25⁄3π × h = 300**
- Multiply by 3: **25π × h = 900**
- Divide by 25π: **h = 900 ÷ (25π) ≈ 11.5 cm**

### Common mistakes

- **Using the slant height:** Make sure you always use the perpendicular height (h) in the formula. Use Pythagoras to find it if necessary.
- **Forgetting the 1⁄3:** A cone has one-third the volume of a cylinder with the same base and height.
- **Using the diameter:** Always halve the diameter first to find the radius.
- **Forgetting to square the radius:** Ensure you calculate r² correctly.