Pythagoras' Theorem

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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 ? Your answer is the 3D-pythagoras formula. 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." Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

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Description of Levels

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.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

Pythagoras' Theorem

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

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Curriculum Reference

Pythagoras' Theorem

Example

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.