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# Surface Area of a Sphere
- URL: https://www.esheets.io/surface-area-of-a-sphere/
- Published: 2026-08-16T12:35:08.000Z
- Updated: 2026-08-16T12:35:08.000Z
- Author: Richard Linnington
- Tags: Geometry and Shape, Maths

This worksheet practises using the formula **4πr²** to find the surface area of a sphere.

You will need to recognise the difference between the radius and the diameter, and apply the method to solid hemispheres to find their total surface area. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This worksheet generates questions on finding the surface area of spheres and solid hemispheres.

The progression moves from straightforward substitution towards reverse problems and scaling factors.

#### What you’ll practise

- Calculating the surface area of a sphere given its radius or diameter.
- Finding exact answers in terms of π.
- Finding the total surface area of a solid hemisphere, including the circular base.
- Solving reverse problems to find the radius when the surface area is known.
- Understanding how changing the radius affects the surface area.

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

## Surface area of a sphere

Work out the surface area of the spheres and hemispheres. For total surface area of a solid hemisphere, remember to include the circular base.

## Topic guide

### The formula for the surface area of a sphere

To find the surface area of a sphere, you use the formula:

**Surface area = 4πr²**

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

### Radius and diameter

Always check whether you have been given the **radius** (from the centre to the edge) or the **diameter** (the full distance across through the centre).

If you are given the diameter, you must divide it by 2 to find the radius before using the formula.

### Worked example: Sphere

*Find the surface area 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: **Surface area = 4 × π × 5²**
- Calculate 5² = 25
- Surface area = 4 × π × 25 = 100π
- Calculate the decimal: 100 × π ≈ 314.159...
- Round to 1 decimal place: **314.2 cm²**

### Total surface area of a solid hemisphere

A hemisphere is exactly half of a sphere. However, a **solid** hemisphere has two parts to its surface:

1. The curved surface (half of a full sphere): **2πr²**
2. The flat circular base: **πr²**

To find the **total** surface area, you must add these together:

**Total surface area = 3πr²**

### Reverse problems

If you are given the surface area and need to find the radius, you set up an equation and solve it.

*The surface area of a sphere is 196π cm². Find its radius.*

- Write the equation: **4πr² = 196π**
- Divide both sides by π: **4r² = 196**
- Divide by 4: **r² = 49**
- Square root: **r = 7 cm**

### Scaling

Because the formula uses **r²**, changing the radius has a squared effect on the surface area.

If you double the radius, the surface area increases by a factor of 2² = 4.  
If you triple the radius, the surface area increases by a factor of 3² = 9.

### Common mistakes

- **Using the diameter instead of the radius:** Always halve the diameter first.
- **Using the volume formula:** Surface area is 4πr², volume is 4/3 πr³.
- **Forgetting to square the radius:** You must calculate r², not just 2r.
- **Wrong units:** Surface area is an area, so it uses square units (e.g. cm², m²), not cubic units.
- **Forgetting the circular base of a hemisphere:** Remember to use 3πr² for the total surface area of a solid hemisphere.
- **Rounding too early:** Keep the full value of π or numbers on your calculator until the final step.