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Pythagoras Theorem

In one line: In any right-angled triangle, the square of the longest side equals the squares of the other two sides added together: a² + b² = c².

Why it works

Pythagoras' theorem lets you work out a missing side of a right-angled triangle (a triangle with one 90° angle, like the corner of a page) without measuring it. Builders use it to check that corners are square, and it also gives the length of a ladder, a screen's diagonal or a shortcut across a field. The longest side, opposite the right angle, is called the hypotenuse (hy-POT-en-yooz) and is always labelled c. 'Squared' means a number multiplied by itself (5² = 5 × 5 = 25), and a 'square root' undoes it (√25 = 5). The theorem is also the starting point for distance on graphs, trigonometry and much of the maths that comes later.

Think of it like: Imagine building a square tile on each side of a right-angled triangle. If the two shorter sides are 3 cm and 4 cm, their tiles have areas of 9 cm² and 16 cm². Cut up those two smaller tiles and they fit exactly onto the tile on the longest side, which has an area of 25 cm². The two small squares always fill the big square perfectly, with nothing left over and no gaps. (Where this breaks: The analogy is about areas, not lengths, so you cannot simply add the side lengths (3 + 4 is 7, not 5); you must square, add, then take the square root. It also only works when the triangle has a right angle; on any other triangle the small squares will not fill the big one exactly.)

Worked example

A rectangular park measures 6 metres by 8 metres. How long is the diagonal path from one corner to the opposite corner?

  1. The diagonal cuts the rectangle into two right-angled triangles. The diagonal is opposite the right angle, so it is the hypotenuse, c.
  2. Square the two shorter sides: 6² = 36 and 8² = 64.
  3. Add them: 36 + 64 = 100, so c² = 100.
  4. Take the square root to undo the squaring: c = √100 = 10.
  5. Check it makes sense: 10 m is longer than either short side (8 m) but shorter than walking around the edges (6 + 8 = 14 m).

Answer: The diagonal path is 10 metres long.

Common mistakes

  • Adding the side lengths instead of their squares (3 + 4 = 7), so always square each side first, then add.
  • Forgetting the square root at the end and giving c² as the answer, so remember the last step is to take √ of your total.
  • Adding when you are finding a shorter side, so when the hypotenuse is already known, subtract instead: a² = c² − b².
  • Labelling the wrong side as c, so always find the side opposite the right angle first, because that is the only side that can be the hypotenuse.

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A right-angled triangle has shorter sides of 3 cm and 4 cm. How long is the hypotenuse?

Questions parents ask

What year do kids learn Pythagoras theorem?

In England it is usually introduced in Year 9, sometimes in Year 8, and revisited in Years 10 and 11. In Australia it is typically taught in Year 9, and in the US it is a Grade 8 topic. Students need to be confident with squaring numbers and square roots first.

How can I help my child with Pythagoras theorem at home?

Use real objects: measure the length and width of a table or phone screen, predict the diagonal with a² + b² = c², then measure it to check. Encourage your child to say the steps aloud: find the hypotenuse, square, add or subtract, then square root. Practising square numbers up to 15² also makes the arithmetic much easier.

Does Pythagoras theorem work for all triangles?

No, it only works for right-angled triangles. However, you can use it in reverse: if a² + b² equals c² for the longest side c, the triangle must have a right angle. This is how builders check that corners are exactly square.

What are Pythagorean triples?

They are sets of three whole numbers that fit a² + b² = c² exactly, such as 3, 4, 5 and 5, 12, 13. Any multiple of a triple also works, so 6, 8, 10 is another one. Spotting them can save time and help children check their answers.