30-60-90 Triangle Calculator

Solve your 30-60-90 triangle

Angle α is fixed at 30° (so β is always 60°). Enter any one remaining value — a leg, the hypotenuse, the altitude, the area, or the perimeter — pre-loaded here with an escalator that rises 15 ft at 30°.

What makes 30-60-90 a "special" right triangle

A 30-60-90 triangle is a right triangle whose acute angles are exactly 30° and 60°. What makes it "special" is that its side lengths always fall in the exact ratio 1 : √3 : 2 — the side opposite the 30° angle (the short leg) is always half the hypotenuse, and the side opposite the 60° angle (the long leg) is always the short leg times √3. That means once you know just one side, the other two follow immediately without needing a calculator for the trigonometry — only a single multiplication or division by √3 or 2.

This calculator still runs the full solver underneath — pinning α to 30° and letting you supply any one of the six remaining quantities (either leg, the hypotenuse, the altitude to the hypotenuse, the area, or the perimeter) — so it works identically whether you're checking a textbook ratio problem or a real measurement that doesn't come out to a clean number.

30-60-90 triangle diagram

Call the short leg x. It faces the 30° angle. The long leg, facing the 60° angle, is x√3, and the hypotenuse is 2x, so the sides are in the ratio 1 : √3 : 2.

<figure>
  <img src="https://righttrianglecalculators.com/images/30-60-90-triangle.png" width="1200" height="800" style="max-width: 100%; height: auto;" alt="A 30-60-90 triangle: the short leg x faces the 30° angle, the long leg is x√3 and the hypotenuse is 2x.">
  <figcaption>30-60-90 triangle side ratios by <a href="https://righttrianglecalculators.com/30-60-90-triangle-calculator">Right Triangle Calculators</a> is licensed under <a href="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>.</figcaption>
</figure>

Worked example: an escalator that climbs 15 ft

Most escalators climb at 30°, so the side profile of one is a 30-60-90 triangle. The rise between floors, here 15 ft, is the short leg, opposite the 30° angle at the bottom. Because the hypotenuse of a 30-60-90 triangle is always twice the short leg, the incline is c = a / sin(30°) = 15 / 0.5 = 30 ft long. The floor distance it covers is the long leg, the short leg times √3: b = 15√3 ≈ 25.98 ft. The sides are 15 : 15√3 : 30, which simplifies to 1 : √3 : 2.

That's useful when planning a building: the escalator needs about 26 ft of floor between the bottom and the top of the incline, plus the landings at each end. From the same triangle, the area of the side profile is (1/2) × 15 × 15√3 ≈ 194.86 ft², the perimeter ≈ 70.98 ft, the altitude to the hypotenuse ≈ 12.99 ft, the inradius ≈ 5.49 ft, and the circumradius is exactly half the hypotenuse, 15 ft.

When you'd use a 30-60-90 triangle

30-60-90 triangles show up constantly in geometry and trigonometry coursework, in equilateral-triangle and hexagon constructions, in roof-pitch and ramp-angle calculations that happen to use a 30° or 60° incline, and in any design that involves bisecting an equilateral triangle. If your triangle instead has a 45° angle, use the 45-45-90 triangle calculator, or switch between both types in the special right triangle calculator. For a general right triangle with any angle at all, use the full right triangle calculator.

30-60-90 triangle FAQ

See the full right triangle FAQ for more general questions.

Cite this page

To cite this calculator in schoolwork or a report, you can use this reference (APA style):

Right Triangle Calculators. (2026, October 10). 30-60-90 triangle calculator. https://righttrianglecalculators.com/30-60-90-triangle-calculator

References

The formulas and definitions on this page match these standard references.

1. Weisstein, Eric W. “30-60-90 Triangle.” MathWorld, Wolfram Research.
2. Weisstein, Eric W. “Equilateral Triangle.” MathWorld, Wolfram Research.
3. Weisstein, Eric W. “Right Triangle.” MathWorld, Wolfram Research.