Special Right Triangles: 30-60-90

The ratio and where each side sits

Sides face their angles: the smallest side is across from the smallest angle. So the short leg $x$ is opposite $30^\circ$, the long leg $x\sqrt{3}$ is opposite $60^\circ$, and the hypotenuse $2x$ is opposite the $90^\circ$ angle.

Two relationships fall straight out of the ratio: the hypotenuse is exactly twice the short leg, and the long leg is the short leg times $\sqrt{3}$. Those two sentences are the whole lesson.

The figure pins each side to its angle: the short leg $x$ faces $30^\circ$, the long leg $x\sqrt{3}$ faces $60^\circ$, and the hypotenuse $2x$ faces the right angle.

Always work through the short leg

Whatever side you are given, find the short leg first, then build the other sides from it. Given the hypotenuse? Divide by $2$. Given the long leg? Divide by $\sqrt{3}$ (and rationalize: $\dfrac{9}{\sqrt{3}} = \dfrac{9\sqrt{3}}{3} = 3\sqrt{3}$). Given the short leg? You are already done with step one.

The classic mistake is doubling the short leg to get the long leg. Doubling gives the hypotenuse. The long leg comes from multiplying by $\sqrt{3}$, which is about $1.73$ — so the long leg is bigger than the short leg but smaller than the hypotenuse, exactly as the ratio says.

How this differs from 45-45-90

A 45-45-90 triangle has two equal legs and ratio $x : x : x\sqrt{2}$ — the only move is multiplying or dividing by $\sqrt{2}$. A 30-60-90 triangle has three different sides, so you must track which leg is which. If you see $\sqrt{2}$ in your work on a 30-60-90 problem, you have mixed up the two triangle types.

Worked examples

Try one yourself

Common questions