# Pythagoras' Theorem

## A self marking exercise on the application of Pythagoras' Theorem.

##### MenuLevel 0Level 1Level 2Level 3Level 4Level 5Level 6Level 7Exam3DHelpMore

Calculate the length of the third side of these right angled triangles. The diagrams are not to scale. Give your answer correct to 1 decimal place.

6.7cm

7cm

cm

9.5cm

8.5cm

cm

9.6cm

11.7cm

cm

6.1cm

8.8cm

cm

8cm

7.2cm

cm

7.2cm

7.1cm

cm

9cm

12.6cm

cm

8.6cm

12.0cm

cm

9.2cm

12.7cm

cm

8.7cm

9.9cm

cm

8.2cm

6.1cm

cm

7cm

11.8cm

cm

7.9cm

12.1cm

cm

7.6cm

6.6cm

cm

8.7cm

11.4cm

cm
Check

This is Pythagoras' Theorem level 3. You can also try:
Level 1 Level 2 Level 4 Level 5 Level 6 Level 7

## Instructions

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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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QI,

Wednesday, June 12, 2019

"The Babylonians were using Pythagoras' Theorem over 1,000 years before Pythagoras was born."

Ann Roberts, London

Thursday, October 1, 2020

"Three D Pythagoras
Suppose you have a cuboid with length l, width w and height h.
Can you find the longest internal length d from one corner to the opposite corner of the box, in terms of l, w and h ?
NOTE: Being able to apply the 2D Pythagoras formula to 3D shapes is still an essential skill, especially if you have a more complex 3D shape."

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## Description of Levels

Close

Level 0 - A 'whole number only' introductory set of questions

Level 1 - Finding the hypotenuse

Level 2 - Finding a shorter side

Level 3 - Mixed questions

Level 4 - Pythagoras coordinates

Level 5 - Mixed exercise

Level 6 - More than one triangle

Level 7 - Harder exercise

Exam Style questions requiring an application of Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles.

Three Dimensions - Three dimensional Pythagoras and trigonometry questions

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

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## Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

## Pythagoras' Theorem

The area of the square on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares on the two shorter sides.

You may have learned the theorem using letters to stand for the lengths of the sides. The corners (vertices) of the right-angled triangle is labelled with capital (upper case) letters. The lengths of the sides opposite them are labelled with the corresponding small (lower case) letters.

Alternatively the sides of the right-angled triangle may me named using the capital letters of the two points they span.

As triangle can be labelled in many different ways it is probably best to remember the theorem by momorising the first diagram above.

To find the longest side (hypotenuse) of a right-angled triangle you square the two shorter sides, add together the results and then find the square root of this total.

To find a shorter side of a right-angled triangle you subtract the square of the other shorter side from the square of the hypotenuse and then find the square root of the answer.

### Example

AB2 = AC2 - BC2
AB2 = 4.72 - 4.12
AB2 = 22.09 - 16.81
AB2 = 5.28
AB = √5.28
AB = 2.3m (to one decimal place)

The diagrams aren't always the same way round. They could be rotated by any angle.

The right-angled triangles could be long and thin or short and not so thin.

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