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# Surface Area of a Cone
- URL: https://www.esheets.io/surface-area-of-a-cone/
- Published: 2026-08-16T14:33:55.000Z
- Updated: 2026-08-16T14:33:55.000Z
- Author: Richard Linnington
- Tags: Geometry and Shape, Maths

This worksheet practises using the formulae for the surface area of a cone.

You will need to calculate the curved surface area (**πrl**) and total surface area (**πrl + πr²**). You will also practise using Pythagoras to find the slant height, and working backwards from a given surface area 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 curved and total surface area of a cone.

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

#### What you’ll practise

- Calculating the curved surface area of a cone.
- Calculating the total surface area of a cone.
- Using Pythagoras to find the slant height when the perpendicular height is given.
- Working backwards to find the slant height or radius when the surface area is known.
- Giving exact answers in terms of π.

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

## Surface area of a cone

Calculate the curved or total surface area of the cones, or work backwards to find a missing length.

## Topic guide

### The formulae

To find the surface area of a cone, you need to know the radius of the base (**r**) and the slant height (**l**).

**Curved surface area = πrl**

**Total surface area = πrl + πr²**

### Radius, perpendicular height and slant height

It is 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.
- **Slant height (l):** The distance along the sloping side of the cone. This is the length used in the surface area formulae.
- **Perpendicular height (h):** The straight vertical distance from the point (apex) to the centre of the base. **Do not use this directly in the surface area formula.**

### Worked example 1: Total surface area

*Find the total surface area of a cone with radius 5 cm and slant height 12 cm. Give your answer to 1 decimal place.*

- Identify the measurements: **r = 5**, **l = 12**
- Calculate the curved surface: **π × 5 × 12 = 60π**
- Calculate the circular base: **π × 5² = 25π**
- Add them together: **60π + 25π = 85π**
- Calculate the decimal: 85 × π ≈ 267.035...
- Round to 1 decimal place: **267.0 cm²**

### Worked example 2: Using Pythagoras

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

*Find the total surface area of a cone with radius 3 cm and perpendicular height 4 cm.*

- Use Pythagoras to find l: **l² = 3² + 4²**
- **l² = 9 + 16 = 25**
- **l = √25 = 5 cm**
- Now use the surface area formula: Total = π × 3 × 5 + π × 3² = 15π + 9π = **24π cm²**

### Reverse problems (working backwards)

You may be given the total surface area and asked to find the slant height or radius.

*Example: The total surface area of a cone is 96π cm². Its radius is 6 cm. Find the slant height.*

- Write the equation: **πrl + πr² = 96π**
- Substitute the radius (r = 6): **6πl + 36π = 96π**
- Divide everything by π: **6l + 36 = 96**
- Subtract 36: **6l = 60**
- Divide by 6: **l = 10 cm**

### Common mistakes

- **Using the perpendicular height:** Make sure you always use the slant height (l) in the formula. Use Pythagoras to find it if necessary.
- **Adding the base incorrectly:** Read the question carefully. Only add the base area (πr²) if it asks for the *total* surface area.
- **Using the diameter:** Always halve the diameter first to find the radius.
- **Wrong units:** Surface area is 2D, so it uses square units (e.g. cm², m²).