The 30-60-90 Triangle

The second special triangle is the 30-60-90 triangle: half an equilateral triangle. Cut an equilateral triangle of side 2 down the middle and you get a right triangle with angles 30°, 60° and 90°, hypotenuse 2 and a short leg of 1.

So the short leg, the one facing the 30° angle, is half the hypotenuse. The long leg, facing 60°, comes from Pythagoras: $\sqrt{{2}^{2}-{1}^{2}} = \sqrt{3}$. The ratio is 1 : √3 : 2, short leg to long leg to hypotenuse.

Here the 30° is at R and the short leg PQ facing it is 4, so the hypotenuse is 8 and the long leg is $4\cdot \sqrt{3}$.