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# Find missing lengths using the cosine rule
- URL: https://www.esheets.io/cosine-rule-missing-lengths/
- Published: 2026-05-01T17:08:42.000Z
- Updated: 2026-07-09T20:28:24.000Z
- Author: Richard Linnington 2
- Tags: Maths, Geometry and Shape

The cosine rule is useful because it lets us find missing lengths in triangles that are not right-angled, where Pythagoras cannot be used. It is especially helpful when we know two sides and the angle between them, and it appears in many real-life problems involving distances, bearings, construction, design and navigation. [Jump to the questions](#practise-now)

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## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise the cosine rule for missing lengths with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Identifying when two sides and the included angle are known.
- Choosing the cosine rule for a missing length.
- Substituting values into the cosine rule.
- Square-rooting to find the missing side.

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

## Topic guide

### What this worksheet practises

This worksheet practises using the cosine rule to find a missing side length in any non-right-angled triangle. You use this rule when you know the lengths of two sides and the size of the angle trapped between them (the included angle).

### Key method

The cosine rule for finding a missing length is:

**a² = b² + c² − 2bc cos(A)**

- 'a' is the missing side you want to find.
- 'A' is the angle opposite side 'a'.
- 'b' and 'c' are the other two known sides.

Substitute your known values into the right-hand side of the equation, calculate the result, and then take the square root to find 'a'.

### Worked example

**Find the length of side x in a triangle where the other two sides are 5 cm and 8 cm, and the angle between them is 60°.**

Step 1: Label the sides and angle. Let a = x, b = 5, c = 8, and A = 60°.

Step 2: Substitute into the formula.

x² = 5² + 8² − (2 × 5 × 8 × cos(60°))

Step 3: Calculate the parts.

x² = 25 + 64 − (80 × 0.5)

x² = 89 − 40

x² = 49

Step 4: Take the square root to find x.

x = √49 = 7 cm

### Common mistakes to avoid

A frequent error is forgetting the final step of taking the square root of the answer. If you calculate a² as 49, the length is not 49 cm; you must take the square root to get 7 cm.

### Things to remember

Ensure your calculator is set to degrees (usually shown as a 'D' or 'Deg' on the screen) before working out trigonometric questions, or your answers will be incorrect.