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Pythagoras' Theorem
Key Facts and Exam Tips
In a right-angled triangle the longest side is called the hypotenuse. It is always opposite the right angle.
Pythagoras’ Theorem states that in a right-angled triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
For the triangle below the theorem can be written as c2 = a2 + b2.
‘Squaring’ a number means multiplying it by itself. So ‘c2’ means ‘c x c’. Remember that c2 represents the area of a square with side c.
When answering questions using Pythagoras’ Theorem, make sure you start with a clear diagram of the triangle. Label the right angle and the sides of the triangle, then fill in the lengths that you know and identify the length you are going to find.
When substituting the values into the formula, take care to replace the letters with the lengths of the correct sides of the triangle. You must be clear about which side is the hypotenuse before you can substitute values.
If the side you need to find is the hypotenuse, you can do the arithmetic straightaway. If you need to find one of the shorter sides, you will need to rearrange the expression first.
At the end of the calculation, you need to take a square root. Make sure you know how to do this on your calculator.
You should know and be able to recognise the first 10 to 15 square numbers, and also the squares of 20, 30, 40 and so on.
You should be familiar with simple Pythagorean triples like (3,4,5), (6,8,10), (5,12,13) and (7,24,25), and know that triangles with sides of these lengths must be right-angled.
Remember to check that the size of your answer is reasonable for the conditions given in the question. If you were asked to find the hypotenuse, your answer should be the biggest of the three sides.
You can put all three values back into the formula and check that they satisfy Pythagoras’ rule. If they don’t, you know you have made a mistake.
Remember to give your answer to the degree of accuracy stated in the question (significant figures, decimal places, or nearest unit of length). If the degree of accuracy is not specified, round your answer appropriately. Round your answer at the end of your calculation.
Use the units given in the question, and be familiar with both metric and imperial systems for units of length.
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