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Pythagoras' Theorem Exercise

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

Level 1 Level 2 Level 3 Level 4 Level 5 Exam-Style Three D Description Help More Pythagoras

Here are some questions which can be answered using Pythagoras' Theorem. You can earn a trophy if you get at least 14 questions correct. Each time you finish a question click the 'Check' button lower down the page to see if you got it right!
[Don't forget to include the units in your answers after question one]

1. What is the name for the longest side of a right angled triangle?

Correct Wrong

2. What is the length of the longest side of a right angled triangle if the two shorter sides are 5cm and 12cm?

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3. Find the length of AB to 1 decimal place.

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4. Find the length of EG to 1 decimal place.

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5. Find the length of JK to 1 decimal place.

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6. A rectangular swimming pool is 23m wide and 44m long. Calculate the length of a diagonal in metres to 1 decimal place.

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7. A ladder is 5m long. How far from the base of a wall should it be placed if it is to reach 4m up the wall? Give your answer in metres correct to 1 decimal place

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8. A tent guy line supports one of the upright tent poles. It runs from the top of the pole to a peg in the ground two and a half metres away from the base of the pole. If the guy line is 363cm long, how tall is the upright tent pole? Give your answer in centimetres correct to the nearest centimetre.

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9. How long is the diagonal of an A4 size sheet of paper? The dimensions of A4 paper are 210 by 297 millimetres (8.3 inches × 11.7 inches). Give your answer in cm to one decimal place.

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10. For international matches football pitches must be of regulation size. The goal lines must be between 64 and 75 metres (70 and 80 yards) long and the touchlines must be between 100 and 110 metres (110 and 120 yards).

What is the difference between the length of the diagonal of the largest acceptable pitch and the length of the diagonal of the smallest acceptable pitch? Give your answer in metres to the nearest metre.

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11. Find the length of a side of a rhombus whose diagonals are of length 14km and 15km. Give your answer in kilometers correct to one decimal place.

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12. Find the perimeter of this parallelogram to one decimal place.

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13. The length of the diagonal of a square is 63m. How long are the sides of the square? Give your answer correct to one decimal place.

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14. Find the height (h) of this isosceles triangle to one decimal place.

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15. The sign says 'Keep off the grass'. Each day Michael has to get from one corner of the rectangular area of grass to the opposite corner. If the park keeper is looking he will walk along the edges but if the park keeper is not looking he will take the direct route, diagonally across the rectangle.

How much further does Michael walk on the days when the park keeper is looking? The length of the rectangular area of grass is 144m and the width is 102m. Give your answer to the nearest metre.

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16. The blue squares have sides of length 26mm and the red square has sides of length 37mm. Find the distance from A to B in centimetres correct to one decimal place.

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17. An irregular quadrilateral ABCD has right angles at the opposite vertices A and C. Calculate the length of the side DA if AB=43.7cm, BC=43.8cm and CD=44.9cm. Give your answers in cm to one decimal place.

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18. An aeroplane flies due north for 263km then changes direction and flies east for 437km. How far is it now in a straight line from its starting position. Give your answer to the nearest kilometre.

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19. A ship sails on a bearing of 045o for 334km then changes direction and sails on a bearing of 135o for 412km. Finally it then turns and sails for 142km on a bearing of 225o. How far is it now in a straight line from its starting position. Give your answer to the nearest kilometre.

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20. One side of a right angled triangle is 10cm. The other two sides are both of length x. Calculate x to 3 significant figures.

Correct Wrong


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


Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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"Very good starters, help pupils settle very well in maths classroom."

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"An absolutely brilliant resource. Only recently been discovered but is used daily with all my classes. It is particularly useful when things can be saved for further use. Thank you!"

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Triangle Solver

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Description of Levels



Level 1 - Finding the hypotenuse

Level 2 - Finding a shorter side

Level 3 - Mixed questions

Level 4 - Pythagoras coordinates

Level 5 - Pythagoras' Theorem 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.

Pythagoras' Theorem

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.

Pythagoras' Theorem

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

Pythagoras' Theorem

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.


Pythagoras 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.

Different proportions

So why is Pythagoras' theorem true?

The video above is from Australia's most unlikely new celebrity, Eddie Woo.

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.