The Pythagorean Theorem

How long a ladder do you need to reach a window, or how big is a television whose screen is “55 inches across the diagonal”? Questions like these are answered by one of the most useful rules in mathematics — Pythagoras' theorem — which ties together the three sides of any right-angled triangle.

A right-angled triangle has one square corner — a right angle. The two sides meeting at that corner are the legs; the longest side, opposite the right angle, is the hypotenuse.

Pythagoras' theorem says that if you build a square on each side, the two smaller squares together cover exactly the same area as the big one:

a^2 + b^2 = c^2

For example, a triangle with legs 3 and 4 has a hypotenuse of 5, because 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

Working backwards: finding a leg

The same equation finds a leg when you already know the hypotenuse and the other leg. Move the known leg across, then undo the square with a square root:

a^2 + b^2 = c^2

Solving for the first leg a:

a^2 = c^2 - b^2 \quad\Longrightarrow\quad a = \sqrt{c^2 - b^2}

and the same move for the other leg b:

b^2 = c^2 - a^2 \quad\Longrightarrow\quad b = \sqrt{c^2 - a^2}

For example, a triangle with hypotenuse 13 and one leg 5 has its other leg at \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12.

See it explained