30-60-90 Triangle Calculator
A 30-60-90 triangle has angles 30°, 60°, and 90° and sides in the fixed ratio 1 : √3 : 2. Enter any one side and get all other sides, the area, and the perimeter instantly.
Known side
Side opposite the 90° right angle; always twice the short leg
- 1
Short leg (given)
5 - 2
Long leg = short leg × √3
5 × √3 = 8.6603 - 3
Hypotenuse = short leg × 2
5 × 2 = 10
How does this calculator work?
A 30-60-90 triangle always has sides in ratio 1 : √3 : 2. If the short leg (opposite 30°) is x, then the long leg is x√3 and the hypotenuse is 2x. Select any known side, enter its length, and get the others plus area (x²√3/2) and perimeter immediately.
Formula
How this is calculated
A 30-60-90 triangle is half of an equilateral triangle sliced along its altitude. Because the angles are fixed, the three sides are always in the ratio 1 : √3 : 2 — the short leg (opposite 30°) to the long leg (opposite 60°) to the hypotenuse (opposite 90°). Once you know any one side, the ratio unlocks all others: if the short leg is x, then the long leg is x√3 and the hypotenuse is 2x. Dividing by √3 or 2 inverts the process for the other starting sides.
The area follows the standard right-triangle formula — half the product of the two legs: Area = ½ × (short leg) × (long leg) = ½ × x × x√3 = (x²√3) / 2. The perimeter is simply the sum of all three sides: x + x√3 + 2x = x(3 + √3).
Because the ratios are exact, results are expressed to four decimal places. The underlying value of √3 ≈ 1.7321, so a short leg of 5 gives a long leg of approximately 8.6603 and a hypotenuse of exactly 10. This calculator covers the geometric relationships only; for general right triangles see the right-triangle solver.
Frequently asked questions
A 30-60-90 triangle is formed by bisecting an equilateral triangle with an altitude. The equilateral triangle has three 60° angles and equal sides; the altitude bisects one angle to 30° and one side to half its length. This forces the three sides into the ratio 1 : √3 : 2 by basic trigonometry (sin 30° = ½, sin 60° = √3/2).
Divide the hypotenuse by 2 to get the short leg x, then multiply x by √3 (≈ 1.7321) to get the long leg. For example, hypotenuse = 10 → short leg = 5 → long leg = 5√3 ≈ 8.660.
Area = ½ × short leg × long leg = ½ × x × x√3 = (x²√3) / 2, where x is the length of the short leg. The two legs serve as the base and height of the right triangle.
Also known as
TG we-Calculate Editorial Team. (2026). 30-60-90 Triangle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-30-60-90-calculator
TG we-Calculate Editorial Team. "30-60-90 Triangle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-30-60-90-calculator.
TG we-Calculate Editorial Team, "30-60-90 Triangle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-30-60-90-calculator
@misc{wecalculate_triangle_30_60_90_calculator, title = {30-60-90 Triangle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-30-60-90-calculator}}, year = {2026}, note = {TG we-Calculate} }
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