What is the 60/30/90 rule?

The ratio

Whether you call it the 60/30/90 or 30-60-90 rule, it is the same triangle: one with angles of 30°, 60°, and 90°. Its sides always come in the fixed ratio

$x : x\sqrt{3} : 2x$

and each side sits opposite a specific angle:

Worked example

If the shortest side is 5, the middle side is $5\sqrt{3}\approx 8.66$ and the hypotenuse is 10. It works in reverse too: a hypotenuse of 12 means the short side is 6 and the middle side is $6\sqrt{3}\approx 10.39$.

Where the rule comes from

Cut an equilateral triangle in half from one corner and you get two 30-60-90 triangles: the hypotenuse is an original side (${2x}$), the short leg is half a side ($x$), and the height follows from the Pythagorean theorem:

$\sqrt{(2x)^2-x^2}=\sqrt{3x^2}=x\sqrt{3}$

You can confirm any 30-60-90 triangle with the calculator on this page: enter the two legs and the hypotenuse should come out to exactly twice the short one.

The other special triangle

The 45-45-90 triangle has its own fixed ratio, $x : x : x\sqrt{2}$: two equal legs, with the hypotenuse a leg times the square root of 2.