Pythagoras’ Theorem

Look for the right angle

A right angle is a quarter turn, 90°, and a drawing marks it with a small square in the corner. A triangle with a right angle in it is called a right triangle.

Pythagoras’ theorem is a rule about the three sides of a right triangle. It works only when one corner is exactly 90°, so the first thing to do is find that corner.

A right triangle whose two sides at the right angle are 3 and 4 long.

The hypotenuse

The side across from the right angle, the one that does not touch it, is called the hypotenuse. It is always the longest side of a right triangle. The two sides that meet at the right angle are the shorter sides, and they are also called the legs.

In this triangle the legs are 3 and 4, and the hypotenuse is the sloping side, marked c. Its length is not given, and the theorem finds it.

The hypotenuse c is the side across from the right angle.

A square on each side

Draw a square on each side of the triangle. The square on the side of 3 has an area of $3^2=3\times 3=9$3² = 3 × 3 = 9 unit squares, and the square on the side of 4 has an area of $4^2=16$4² = 16.

The theorem says that the two smaller squares together have the same area as the square on the hypotenuse. So the square on the hypotenuse has an area of 9 + 16 = 25. The side of a square of area 25 is 5, because $5^2=25$5² = 25, so the hypotenuse is 5.

Sides of 3 and 4 meet at a corner that turns, with a square on every side. At 60° the square on the third side is smaller than 9 + 16 = 25: its side is only about 3.6. Drag the corner to 90°: the square on the third side is then exactly 9 + 16 = 25. The button under it shows the unit tiles, so that 9, 16 and 25 can be counted.

The rule for every right triangle

Call the two legs a and b and the hypotenuse c. Then Pythagoras’ theorem says that in every right triangle, $a^2+b^2=c^2$a² + b² = c².

To find the hypotenuse, square the two legs, add the squares, and take the square root. If the legs are 6 cm and 8 cm, then $c^2=6^2+8^2=36+64=100$c² = 6² + 8² = 36 + 64 = 100, so $c=\sqrt{100}=10$c = √100 = 10 cm. Check: 10 × 10 = 100.

The answer is not always a whole number. If the legs are 2 cm and 3 cm, then $c^2=4+9=13$c² = 4 + 9 = 13, so $c=\sqrt{13}$c = √13, which is 3.61 cm to 2 decimal places.

The legs are a and b and the hypotenuse is c, so $a^2+b^2=c^2$a² + b² = c².

Only with a right angle

A triangle with sides 4, 5 and 6 has no right angle, and the rule fails: $4^2+5^2=16+25=41$4² + 5² = 16 + 25 = 41, but $6^2=36$6² = 36. The same happens to the sides 3 and 4 when the corner between them is not 90°: the square on the third side is not 9 + 16.

The rule also explains why the hypotenuse is the longest side. $c^2$c² is $a^2$a² with $b^2$b² added on, so $c^2$c² is more than $a^2$a², and c is longer than a. In the same way c is longer than b. If a hypotenuse you work out comes out shorter than one of the legs, a step has gone wrong.

The usual mistakes

Adding the sides instead of their squares. 3 + 4 = 7 is the distance along the two legs, walking around the corner. The hypotenuse goes straight across, so it is shorter: 5.

Stopping at the square. $c^2=25$c² = 25 is the area of the square on the hypotenuse, not its length. Take the square root: c = 5.

Adding the squares of the wrong sides. The two squares that are added belong to the legs, the sides that meet at the right angle. The square on the hypotenuse is their total, never one of the two that are added.