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Pythagoras theorem

In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two sides.

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The big idea

Builders, ladders and map makers all use one famous rule about right-angled triangles. It lets you find a missing side without measuring it.

A right-angled triangle has one angle of 90°. The side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called the legs.

The Pythagoras theorem says: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the legs are a and b and the hypotenuse is c, then c² = a² + b².

For the 3, 4, 5 triangle: 3² + 4² = 9 + 16 = 25 = 5². Picture a square drawn on each side. The two smaller squares together have the same area as the big square.

The rule also works backwards. If the sides of a triangle satisfy c² = a² + b², then the triangle has a right angle. Sets of whole numbers like 3, 4, 5 or 5, 12, 13 that fit the rule are called Pythagorean triples.

Squares on the sides of a 3-4-5 triangleA right-angled triangle with legs 3 and 4 and hypotenuse 5, with a square drawn on each side. The square on the leg of 3 has area 9, the square on the leg of 4 has area 16, and the square on the hypotenuse has area 25. 9 plus 16 equals 25. 3² = 9 4² = 16 5² = 25 3² + 4² = 5² 9 + 16 = 25 c² = a² + b² c is the longest side
The areas of the two small squares, 9 and 16, add up to the area of the big square, 25.

Worked examples

See how it works

  1. Find the hypotenuse

    The legs are 6 cm and 8 cm. c² = 6² + 8² = 36 + 64 = 100. So c = √100 = 10 cm.

  2. Find a leg

    The hypotenuse is 13 cm and one leg is 5 cm. The other leg b satisfies b² = 13² − 5² = 169 − 25 = 144. So b = √144 = 12 cm.

  3. A ladder against a wall

    A 10 m ladder has its foot 6 m from the wall. The height it reaches is h, where h² = 10² − 6² = 100 − 36 = 64. So h = 8 m.

  4. Is it a right triangle?

    The sides are 9, 12 and 15. The longest is 15. 9² + 12² = 81 + 144 = 225 and 15² = 225. They are equal, so yes, it is a right-angled triangle.

Try it at home

The knotted-string triangle

You will need

  • A piece of string about 60 cm long
  • A ruler
  • A pen
  • A book or notebook with a square corner

What to do

  1. Mark a point on the string every 5 cm so that there are 12 equal parts.
  2. Lay the string on a table in a triangle with sides of 3 parts, 4 parts and 5 parts. Hold the corners in place with your fingers.
  3. Place the corner of the book in the corner between the 3 side and the 4 side.
  4. Check: it fits exactly, so the angle is 90°. This is because 3² + 4² = 5².
  5. Try sides of 3, 3 and 6 parts or 2, 5 and 5 parts. Do they make a square corner?

Ask a grown-up: A grown-up can hold the corners while you measure. No sharp tools are needed.

Fun fact

Ancient Indian texts called the Sulba Sutras, written more than 2,000 years ago, already describe this rule for building altars.

Practice sheet

Check what you learned

Write your answers in a notebook, or print this sheet and write on it. When you finish, mark your work with the answer key.

Practice sheet: Pythagoras theorem (Classes 9-10, Maths)

  1. What is the longest side of a right-angled triangle called?
  2. The legs of a right triangle are 3 cm and 4 cm. Find the hypotenuse.
  3. The legs are 5 cm and 12 cm. Find the hypotenuse.
  4. The legs are 9 cm and 12 cm. Find the hypotenuse.
  5. The hypotenuse is 17 cm and one leg is 8 cm. Find the other leg.
  6. Is a triangle with sides 6 cm, 8 cm and 10 cm right-angled?
  7. Is a triangle with sides 4 cm, 5 cm and 6 cm right-angled?
  8. A rectangle is 24 cm long and 7 cm wide. Find the length of its diagonal.
  9. A 5 m ladder has its foot 3 m from a wall. How high up the wall does it reach?
  10. Ravi walks 12 m west and then 5 m north. How far is he from where he started, in a straight line?
  11. A square has side 10 cm. Find its diagonal, correct to one decimal place (take √2 = 1.414).

Answer key

Answers: Pythagoras theorem

Do the practice sheet first, then check here. If an answer is different from yours, read the lesson again and try once more.

  1. The hypotenuse.
  2. 5 cm (9 + 16 = 25).
  3. 13 cm (25 + 144 = 169).
  4. 15 cm (81 + 144 = 225).
  5. 15 cm (289 − 64 = 225, and √225 = 15).
  6. Yes. 36 + 64 = 100 = 10².
  7. No. 16 + 25 = 41, but 6² = 36.
  8. 25 cm (576 + 49 = 625).
  9. 4 m (25 − 9 = 16).
  10. 13 m (144 + 25 = 169).
  11. About 14.1 cm (diagonal = √(100 + 100) = √200 = 10√2 = 14.14).

What next

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