Pythagorean theorem

What the symbols mean

$a$

one of the two shorter sides (legs), which meet at the right angle

$b$

the other shorter side

$c$

the hypotenuse: the longest side, opposite the right angle

When to use it

Use it whenever a triangle has a right angle and you know two of its sides: the length of a ladder against a wall, the diagonal of a screen or a field, the straight-line distance between two points on a grid. It only works for right triangles; for any other triangle use the law of cosines, which reduces to Pythagoras when the angle is 90°.

Why it works

The graph shows the classic picture: build a square on each side. The two smaller squares together have exactly the same area as the big one. One proof arranges four copies of the triangle inside a square of side $a+b$ in two different ways; the space left over is $c^2$ in one arrangement and $a^2+b^2$ in the other.

Common mistakes

• Adding when you should subtract. When the unknown is a shorter side, $a^2=c^2-b^2$; the hypotenuse is always the longest side.
• Forgetting the square root at the end: $a^2=64$ means $a=8$, not 64.
• Writing $\sqrt{a^2+b^2}=a+b$. The square root of a sum is not the sum of the square roots: $\sqrt{3^2+4^2}=5$, not 7.
• Using it on a triangle without a right angle.