30-60-90 triangle

A 30-60-90 triangle is a special right triangle whose angles measure 30°, 60°, and 90°. Its defining property is a fixed side ratio: the sides are always in the proportion x : x√3 : 2x. The shortest side (x) sits opposite the 30° angle, the longer leg (x√3) sits opposite the 60° angle, and the hypotenuse (2x) sits opposite the right angle.

The ratio makes missing-side problems fast. If the short leg is 5, the long leg is 5√3 and the hypotenuse is 10. Working backward takes one extra step: if the hypotenuse is 12, the short leg is 6 and the long leg is 6√3; if the long leg is 9, divide by √3 to get a short leg of 9/√3 = 3√3, making the hypotenuse 6√3. The most common mistake is attaching √3 to the wrong side — remember, the √3 belongs to the leg opposite 60°, never to the hypotenuse.

This triangle appears constantly because it is half of an equilateral triangle: dropping an altitude from one vertex of an equilateral triangle splits it into two 30-60-90 triangles. That connection is the source of many area and height problems involving equilateral triangles and regular hexagons.

The 30-60-90 side ratios are heavily tested on the GRE, ACT, and CLT math sections, alongside the other special right triangle, the 45-45-90. Memorize the ratio cold — test questions are built so that recognizing the pattern replaces slower Pythagorean theorem or trigonometry calculations.