30-60-90 Triangle Calculator

Enter any one side of a 30-60-90 special right triangle to find all sides, area, and perimeter.

Enter a Known Side

Result

All Three Sides
5 : 8.6603 : 10
units
Short leg (a) - opposite 30°5
Long leg (b) - opposite 60°8.660254
Hypotenuse (c) - opposite 90°10
Area21.650635
Perimeter23.660254

Step-by-Step Solution

Ratio: 1 : sqrt(3) : 2 => 5 : 8.6603 : 10

The 30-60-90 Special Right Triangle

A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle are always in the fixed ratio 1 : √3 : 2. This means if the shortest side (opposite the 30° angle) has length a, then the longer leg is a√3 and the hypotenuse is 2a.

Side Ratios

The three sides are always proportional to 1, √3, and 2.

a : a*sqrt(3) : 2a

Area Formula

Using the two legs as base and height.

A = (a^2 * sqrt(3)) / 2

Perimeter

Sum of all three sides.

P = a + a*sqrt(3) + 2a = a(3 + sqrt(3))

How to Solve a 30-60-90 Triangle

You only need one side to find all three. Identify which side you know (short leg, long leg, or hypotenuse) and use the ratio to find the others:

  • Given the short leg (a): Long leg = a√3, Hypotenuse = 2a
  • Given the long leg (b): Short leg = b/√3, Hypotenuse = 2b/√3
  • Given the hypotenuse (c): Short leg = c/2, Long leg = c√3/2

Real-World Applications

30-60-90 triangles appear frequently in architecture, engineering, and design. They are half of an equilateral triangle and are used in roof pitch calculations, hexagonal structures, and trigonometric computations. The fixed ratios make mental math possible without a calculator for common problems.