The 30-60-90 Triangle

The ratio comes from cutting an equilateral triangle in half. Start with an equilateral triangle with all sides length 2x and all angles 60 degrees. Drop a line straight down from the top vertex to the midpoint of the base. That line splits the 60 degree top angle into two 30 degree angles, and splits the base into two pieces of length x each. One half is now a right triangle: hypotenuse 2x, shorter leg x, and the bisecting line itself as the longer leg. The Pythagorean theorem turns up the length of that line: (2x)² = x² + (longer leg)², which simplifies to longer leg = x root 3. Every 30-60-90 triangle is a copy of this half-triangle, so the ratio x : x root 3 : 2x never changes, only the scale does.

Because the ratio is fixed, one known side unlocks the whole triangle without a trig table. If you're given the shorter leg, multiply by root 3 for the longer leg and by 2 for the hypotenuse. If you're handed the hypotenuse instead, reverse the first step: divide by 2 to get the shorter leg, then multiply that by root 3 to get the longer leg.

This shortcut sits next to the 45-45-90 triangle as the two special right triangles worth memorizing cold, and it's really a shortcut version of the sine and cosine ratios for 30 and 60 degrees. Once trig ratios and the Law of Sines or Law of Cosines come into play for non-right triangles, the 30-60-90 pattern often shows up hiding inside a larger figure, so spotting it fast saves real time.