Special Right Triangles: 30-60-90

Core Idea

A 30-60-90 triangle has sides in the ratio 1 : sqrt(3) : 2. The shortest side (opposite 30) is half the hypotenuse, and the longer leg (opposite 60) is sqrt(3) times the shorter leg. This triangle arises from bisecting an equilateral triangle. Knowing these ratios allows exact computation without a calculator and provides the exact values for sin/cos/tan of 30 and 60 degrees.

How It's Best Learned

Derive the ratios by cutting an equilateral triangle in half and applying the Pythagorean theorem to find the altitude. Practice scaling: if the hypotenuse is 10, the short leg is 5 and the long leg is 5*sqrt(3). Work problems in both directions (given any one side, find the others).

Common Misconceptions

• Putting sqrt(3) with the wrong side (it goes with the longer leg, opposite 60, not the shorter leg).
• Confusing 30-60-90 ratios with 45-45-90 ratios.
• Thinking the ratio 1:sqrt(3):2 means the sides are literally 1, sqrt(3), and 2 rather than being scalable by any factor.

Explainer

You know the Pythagorean theorem: in a right triangle, a² + b² = c². The 30-60-90 triangle gives you a specific set of side ratios you can derive once and then use forever — no calculator required. Start with an equilateral triangle where every side has length 2 and every angle is 60°. Cut it straight down the middle from one vertex to the opposite side's midpoint. This produces two congruent right triangles. Each right triangle has a hypotenuse of 2 (one full side of the equilateral triangle), a short leg of 1 (half the base), and angles of 30°, 60°, and 90°. The long leg — the altitude you just cut along — can be found with the Pythagorean theorem: 1² + h² = 2², so h² = 3, and h = √3. The sides are in ratio 1 : √3 : 2.

The mnemonic for which side goes where: the short leg (length 1) is opposite the 30° angle, the long leg (length √3) is opposite the 60° angle, and the hypotenuse (length 2) is opposite the 90° angle. Opposite the biggest angle is the longest side — that ordering is consistent with everything you know about triangles. The √3 always belongs to the 60° side; many mistakes come from accidentally swapping the two legs.

These ratios are scalable. If the hypotenuse is 10, multiply every ratio value by 5: short leg = 5, long leg = 5√3. If the short leg is 7, the hypotenuse is 14 and the long leg is 7√3. The scale factor is whatever you need to match the given side. The only rule: identify which side you're given and which position (short leg / long leg / hypotenuse) it occupies, then compute the scale factor from there.

The payoff extends into trigonometry. The sine, cosine, and tangent of 30° and 60° come directly from this triangle. sin(30°) = opposite/hypotenuse = 1/2; cos(30°) = adjacent/hypotenuse = √3/2; tan(30°) = 1/√3. For 60°, the roles of the legs swap: sin(60°) = √3/2; cos(60°) = 1/2; tan(60°) = √3. These are the exact values you'll use throughout trigonometry and precalculus — knowing their origin in the 30-60-90 triangle makes them impossible to forget and easy to re-derive when needed.