Pythagoras’ Theorem

For a right-angled triangle with hypotenuse $c$ and shorter sides $a$ and $b$: $a^2+b^2=c^2$.

$\text{Finding the hypotenuse}$: square the two shorter sides, add, then square root.

$\text{Finding a shorter side}$: square the hypotenuse, subtract the square of the known shorter side, then square root.

The converse of Pythagoras is also true: if the side lengths satisfy $a^2+b^2=c^2$, the triangle has a right angle opposite $c$.

Triples of whole numbers that work, like $3,4,5$ and $5,12,13$, are called Pythagorean triples. Multiples also work: $6,8,10$.

The line between two points is the hypotenuse of a right-angled triangle whose other sides are the horizontal and vertical differences.

So the distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.

To find the diagonal of a cuboid, use Pythagoras twice: first the diagonal of the base, then the triangle formed by that diagonal and the vertical edge.

Combined, the space diagonal of a cuboid with sides $a$, $b$, $c$ is $\sqrt{a^2+b^2+c^2}$.