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# Pythagoras in surd form
- URL: https://www.esheets.io/pythagoras-in-surd-form/
- Published: 2025-03-27T23:02:28.000Z
- Updated: 2026-06-21T16:47:03.000Z
- Author: Richard Linnington
- Tags: Maths, Number, Geometry and Shape

When calculating the distance between two points – whether it’s across a field, up a ladder, or even in a video game – Pythagoras’ Theorem is your go-to tool. But sometimes the answer isn’t a neat whole number. That’s where surds come in: they let us leave square roots in their exact form, keeping your answers precise without reaching for a calculator. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise Pythagoras in surd form 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

- Using Pythagoras’ theorem exactly.
- Leaving the square root in surd form.
- Simplifying surds where possible.
- Avoiding unnecessary decimal rounding.

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

Give your answer in the form **a√b** and in its **simplest form**.

  
## Topic guide

### What this worksheet practises

This worksheet provides practice on using Pythagoras' Theorem where the final answer must be left as a "surd" (an exact square root like √45) rather than a rounded decimal. This is extremely common in non-calculator exams.

### Key method

The Pythagoras method is exactly the same, you just skip the very last calculator step and simplify the surd instead.

- Identify your sides: 'a' and 'b' are the short sides, 'c' is the hypotenuse (the longest side, opposite the right angle).
- Use the formula: **a² + b² = c²** (to find the hypotenuse) or **c² − b² = a²** (to find a short side).
- Once you have a value for a² or c², put a square root symbol over the number. E.g. c = √50.
- **Simplify the Surd:** Look for the largest square number (4, 9, 16, 25, 36...) that divides exactly into your number. Split the root and simplify it.

### Worked example

**A right-angled triangle has short sides of 5cm and 5cm. Find the exact length of the hypotenuse in simplified surd form.**

Step 1: Set up the formula. We need 'c'.

5² + 5² = c²

25 + 25 = c²

50 = c²

Step 2: Put it in a root.

c = √50.

Step 3: Simplify the surd. The largest square number that goes into 50 is 25.

√50 = √(25 × 2)

√50 = √25 × √2

c = 5√2.

The exact length is 5√2 cm.

### Common mistakes to avoid

A common error is stopping at √50 and failing to simplify it to 5√2\. If the question specifically asks for "simplified surd form" or "the form a√b", stopping early will cost you the final method mark.

### Things to remember

If you perform your calculation and get a number like c² = 17, and you cannot find any square number that divides into 17 (because 17 is prime), then your final answer is simply √17\. Not all surds can be simplified.