30-60-90 Triangle Calculator

To solve a 30-60-90 triangle, you only need one side length. If the short leg (opposite 30°) is 'a', then the long leg equals a√3 and the hypotenuse equals 2a. Simply identify which side you know and apply the ratio 1 : √3 : 2 to find the rest.

The key rule is that the sides are always in the ratio 1 : √3 : 2, corresponding to the sides opposite 30°, 60°, and 90° respectively. This means the hypotenuse is always twice the shortest leg, and the longer leg is always the short leg multiplied by √3.

The side ratios are 1 : √3 : 2. If the short leg is 1 unit, the long leg is approximately 1.732 units and the hypotenuse is 2 units. These ratios are fixed regardless of the actual size of the triangle.

If the hypotenuse (c) is known, divide it by 2 to get the short leg: a = c / 2. Then multiply the short leg by √3 to get the long leg: b = (c / 2) × √3. For example, if c = 10, then a = 5 and b = 5√3 ≈ 8.660.

With a hypotenuse of 10, the short leg a = 10 / 2 = 5 and the long leg b = 5√3 ≈ 8.660. The area = (1/2) × a × b = (1/2) × 5 × 5√3 = 12.5√3 ≈ 21.651 square units.

It's called special because its angles are always exactly 30°, 60°, and 90°, and its side lengths always follow the fixed ratio 1 : √3 : 2. This predictable relationship means you can solve the entire triangle from just one known side, without needing trigonometric tables.

The area is calculated using the two legs as base and height: Area = (1/2) × a × b, where a is the short leg and b is the long leg. Since b = a√3, this simplifies to Area = (a² × √3) / 2.

The perimeter is the sum of all three sides: P = a + b + c = a + a√3 + 2a = a(3 + √3). If the short leg is 5 units, the perimeter = 5 × (3 + √3) ≈ 5 × 4.732 ≈ 23.66 units.