Pythagoras' theorem, with examples

In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. If the legs are a and b and the hypotenuse is c:

a² + b² = c²

The theorem carries the name of the Greek thinker Pythagoras, but the rule was known earlier in several places. In India it is stated in the Baudhayana Sulbasutra, a text on building fire altars that is centuries older.

Finding the hypotenuse

Legs 6 cm and 8 cm: c² = 36 + 64 = 100, so c = 10 cm. Square both legs, add, then take the square root.

Finding a leg

Hypotenuse 13 cm and one leg 5 cm: the other leg squared is 169 − 25 = 144, so it is 12 cm. Here you subtract, because the hypotenuse is the largest side.

Pythagorean triples

Whole numbers that fit the theorem are called Pythagorean triples. Knowing a few saves time.

Any multiple of a triple is also a triple: 6, 8, 10 and 9, 12, 15 come from 3, 4, 5.

The converse: testing for a right angle

If the three sides of a triangle satisfy a² + b² = c², the triangle is right-angled, with the right angle opposite the longest side. Sides 7, 24, 25: 49 + 576 = 625 = 25². It is right-angled. Sides 5, 6, 8: 25 + 36 = 61, but 64 is larger, so the angle is more than 90° and the triangle is obtuse.

Masons and carpenters use this. A rope knotted into lengths of 3, 4 and 5 units, pulled tight into a triangle, gives a true right angle for laying out a wall.

Where it is used

• The diagonal of a rectangle: a screen 16 units wide and 9 high has a diagonal of √(256 + 81) = √337, about 18.36 units.
• A ladder against a wall: a 5 m ladder with its foot 3 m from the wall reaches 4 m up.
• The height of an equilateral or isosceles triangle, by splitting it down the middle.
• The slant height of a cone or pyramid, from its height and radius.
• Distance between two points on a graph: the differences in x and y are the two legs.

Why it is true: a picture proof

Take four copies of the triangle and arrange them inside a square of side a + b so that their hypotenuses form a tilted square of side c in the middle. The big square has area (a + b)². It is also four triangles plus the middle square: 4 × ½ab + c². Setting them equal, a² + 2ab + b² = 2ab + c², and the 2ab cancels to leave a² + b² = c².

Common mistakes

• Adding the sides before squaring: √(6 + 8) is not the hypotenuse.
• Adding when you should subtract, when the hypotenuse is given.
• Using the theorem in a triangle that has no right angle.
• Forgetting the square root at the end, and giving c² as the answer.

See the right-angled triangle page, or use the calculator.

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