Pythagoras

Explanation

In a right triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle. The three squares on the sides show this as areas: with a 3-4-5 triangle, 9 + 16 = 25.

Want the calculation with steps? Open the Pythagorean theorem calculator.

Which side is which?

The two sides that form the right angle are the legs, a and b. In Dutch schools they are called rechthoekszijden. The side across from the right angle is the hypotenuse c (schuine zijde), and it is always the longest one. Find the little square that marks the right angle first; the side that does not touch it is c.

Why is it true?

Set the proof step in the simulation to 1. Four identical triangles sit inside a square with side a + b, leaving a tilted square of area c² in the middle. Step 2 moves the same four triangles, and now the empty space is two squares, a² and b². Nothing about the big square or the triangles changed, so the empty space must be the same size both times: c² = a² + b².

Examples by level

• vmbo: a 3,5 m ladder has its foot 1 m from a wall. The ladder is the hypotenuse, so you subtract: 3,5² − 1² = 12,25 − 1 = 11,25. It reaches √11,25 ≈ 3,35 m up the wall.
• havo: legs of 5 and 12. Then 5² + 12² = 25 + 144 = 169, so c = √169 = 13.
• vwo: the distance from (−1; 2) to (5; 10). The steps are 6 across and 8 up, so the distance is √(6² + 8²) = √100 = 10.

When does it not work?

Only triangles with a right angle follow the rule. Turned around, it tells you whether an angle is right: for sides 7, 8 and 9 you get 7² + 8² = 49 + 64 = 113, but 9² = 81. The two do not match, so there is no right angle in that triangle.

Tasks

1. Make the hypotenuse c = 10 with a = 6.
2. Set a = 5 and choose b so that the hypotenuse is 13 long.
3. Make both legs equally long and the square on the hypotenuse exactly 50.

Common mistakes

Test yourself

Practice with Pythagoras practice.

Work it out

Pythagorean theorem calculator