The Theorem of Pythagoras

≈ 25 min

In a right-angled triangle, the side opposite the right angle is the longest and is called the hypotenuse. The Theorem of Pythagoras connects the three sides:

a^2 + b^2 = c^2

where $c$ is the hypotenuse and $a, b$ are the two shorter sides (the legs).

To find the hypotenuse, square the two legs, add them, then take the square root:

c = \sqrt{a^2 + b^2}

Worked example. Legs of $3$ and $4$ . Then $c^2 = 3^2 + 4^2 = 9 + 16 = 25$ , so $c = \sqrt{25} = 5$ .

In $a^2 + b^2 = c^2$ , the side $c$ is…

the shortest side

the hypotenuse (longest side, opposite the right angle)

any side you choose

the right angle itself

A right-angled triangle has legs $3$ and $4$ . Find the hypotenuse.

A right-angled triangle has legs $6$ and $8$ . Find the hypotenuse.

A right-angled triangle has legs $5$ and $12$ . Find the hypotenuse.