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# Pythagoras and isosceles triangles
- URL: https://www.esheets.io/pythagoras-and-isosceles-triangles/
- Published: 2025-11-09T13:34:05.000Z
- Updated: 2026-06-21T17:37:55.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Architects and engineers often use isosceles triangles when designing roofs, bridges, and supports. Pythagoras' Theorem helps calculate missing lengths—like the slant height of a roof—when only the base and height are known. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise Pythagoras and isosceles triangles 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

- Recognising the isosceles triangle structure.
- Splitting the triangle where needed.
- Using Pythagoras’ theorem on a right-angled triangle.
- Finding the missing side or height.

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

Calculate the missing lengths for these isosceles triangles. **All answers should be rounded to one decimal place.** 

## Topic guide

### What this worksheet practises

This worksheet focuses on applying Pythagoras' Theorem to solve problems involving isosceles triangles. Pythagoras' Theorem (a² + b² = c²) only works on right-angled triangles. An isosceles triangle does not have a right angle, so you must create one first.

### Key method

The secret is to slice the isosceles triangle in half.

- Draw a straight vertical line from the top point (the apex) straight down to the middle of the base.
- This line represents the **height** of the triangle. It cuts the isosceles triangle into two perfectly identical right-angled triangles.
- **Crucial Step:** Halve the length of the original base. This gives you the base length for your new right-angled triangle.
- Use Pythagoras' Theorem on this new right-angled triangle to find either the height or the sloping side.

### Worked example

**An isosceles triangle has a base of 10cm and two equal sloping sides of 13cm. Find the height of the triangle.**

Step 1: Cut the triangle in half. The base of our new right-angled triangle is half of 10.

New base = 5cm.

Step 2: Identify the sides of the right-angled triangle. We have the base (a=5) and the hypotenuse (c=13). We need to find the height (b).

Step 3: Set up Pythagoras.

5² + b² = 13²

25 + b² = 169

Step 4: Solve for b.

b² = 169 − 25

b² = 144

b = √144 = 12.

The height of the triangle is 12cm.

### Common mistakes to avoid

The most devastating mistake is forgetting to halve the base before starting Pythagoras. If a student uses the full base of 10cm alongside the hypotenuse of 13cm, the calculation will be completely wrong. Always remember: Pythagoras only works on right angles.

### How to check your answer

The height of an isosceles triangle must always be slightly shorter than the sloping sides. In our example, a height of 12cm is mathematically sensible compared to the sloping side of 13cm. If you calculated a height of 15cm, you would instantly know an error was made.