> ## Content Index
> Fetch the complete content index at: https://www.esheets.io/llms.txt
> Use this file to discover other available public pages before exploring further.

# Volume of a Sphere
- URL: https://www.esheets.io/volume-of-a-sphere/
- Published: 2026-08-16T12:30:56.000Z
- Updated: 2026-08-16T12:30:56.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

This worksheet practises using the formula **4/3 πr³** to find the volume of a sphere.

You will need to recognise the difference between the radius and the diameter, find the volume of hemispheres, and work backwards from a given volume to find the radius. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This worksheet generates questions on finding the volume of spheres and hemispheres.

The progression moves from straightforward substitution towards finding the radius from a given volume, and finishes with an applied problem.

#### What you’ll practise

- Calculating the volume of a sphere given its radius or diameter.
- Finding exact volumes in terms of π.
- Finding the volume of a hemisphere.
- Working backwards to find the radius when the volume is known.
- Equating the volume of a sphere to another shape to solve a problem.

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

## Volume of a sphere

Work out the volume of the spheres and hemispheres, or use the volume to find the radius.

## Topic guide

### The formula for the volume of a sphere

To find the volume of a sphere, use the formula:

**Volume = 4/3 πr³**

where **r** is the radius of the sphere.

### Radius and diameter

Make sure you have been given the **radius** (from the centre to the edge). If you are given the **diameter** (the full distance across), you must divide it by 2 to find the radius before using the formula.

### Worked example: Sphere

*Find the volume of a sphere with a radius of 5 cm. Give your answer to 1 decimal place.*

- Identify the radius: **r = 5**
- Substitute into the formula: **Volume = 4/3 × π × 5³**
- Calculate 5³ = 125
- Volume = 4/3 × π × 125 = 500/3 π
- Calculate the decimal: 500 ÷ 3 × π ≈ 523.598...
- Round to 1 decimal place: **523.6 cm³**

### Volume of a hemisphere

A hemisphere is half of a sphere. To find its volume, simply halve the formula:

**Volume = 2/3 πr³**

Unlike surface area, you do not need to add anything for the flat circular base when finding the volume.

### Exact answers in terms of π

Sometimes you will be asked to give your answer in terms of π. This means you multiply the numbers together but leave π as a symbol.

*Example: Find the volume of a sphere with a radius of 3 cm. Give your answer in terms of π.*

- Substitute: **Volume = 4/3 × π × 3³**
- Calculate 3³ = 27
- Multiply the numbers: 4/3 × 27 = 36
- Final answer: **36π cm³**

### Reverse problems (working backwards)

If you are given the volume and need to find the radius, you can set up an equation and solve it using a cube root.

*Example: The volume of a sphere is 1500 cm³. Find its radius to 1 decimal place.*

- Write the equation: **4/3 πr³ = 1500**
- Multiply by 3 and divide by 4: **πr³ = 1125**
- Divide by π: **r³ = 1125 / π** (approx 358.09)
- Cube root: **r = ∛358.09 ≈ 7.1 cm**

### Conservation of volume

In some applied problems, one shape is melted down and turned into another. The key is that the **volume remains the same**.

If a cuboid is melted to form a sphere, calculate the volume of the cuboid first. Then, set the volume of the sphere equal to that number and solve for the radius just like a normal reverse problem.

### Common mistakes

- **Using the surface area formula:** Remember volume uses 4/3 πr³, not 4πr².
- **Forgetting to cube the radius:** Ensure you calculate r³ (r × r × r), not r² or 3r.
- **Using the diameter:** Always halve the diameter first to find the radius.
- **Wrong units:** Volume is 3D, so it uses cubic units (e.g. cm³, m³), not square units.
- **Rounding too early:** Do not round your numbers until the very final step of the calculation.