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30-60-90 Triangle Calculator

Enter one side and select which side it is to solve the 30-60-90 triangle.

How it is calculated

You give one positive side length and say whether it is the short leg (opposite 30°), the long leg (opposite 60°) or the hypotenuse. The tool first finds the short leg a: a = value, or a = value ÷ √3 for the long leg, or a = value ÷ 2 for the hypotenuse. Then long leg b = a√3, hypotenuse c = 2a, area ½ × a × b and perimeter a + b + c. All results have four decimals and use the same unit as your input (area in that unit squared). Zero or negative lengths give no result.

Worked example

The long leg is 10. Short leg: 10 ÷ 1.7320508 = 5.7735. Hypotenuse: 2 × 5.7735 = 11.5470. Area: ½ × 5.7735 × 10 = 28.8675. Perimeter: 5.7735 + 10 + 11.5470 = 27.3205. Check with the Pythagorean theorem: 5.7735² + 10² = 33.333 + 100 = 133.333, and 11.5470² = 133.333. If instead 10 were the hypotenuse, the short leg would be 5, the long leg 8.6603 and the area 21.6506.

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Common mistakes

  • Choosing the wrong side type: 10 as the long leg gives a short leg of 5.7735, while 10 as the short leg gives a hypotenuse of 20.
  • Assuming the side opposite 60° is the middle number 2: the ratio is 1 : √3 : 2, so the long leg is 1.732 times the short leg, not twice.
  • Using the tool for a triangle that is not 30-60-90: other right triangles, such as 45-45-90 or 3-4-5, have different ratios.
  • Treating the hypotenuse as a leg: the hypotenuse is always the longest side and is exactly double the shortest.
  • Mixing units in area: the area is in square units, so 10 cm sides give an area in cm², not cm.

About this calculator

A 30-60-90 triangle is a special right triangle whose angles are fixed at 30°, 60°, and 90°, which forces its three sides into a constant ratio of 1 : √3 : 2 — the side opposite 30° is always the shortest, the side opposite 60° is √3 times that length, and the hypotenuse (opposite the right angle) is exactly twice the shortest side. Because the ratio never changes, knowing the length of just one side is enough to derive the other two, along with the triangle's area and perimeter.

This calculator takes any one known side, lets you specify which of the three it is, and solves for the remaining sides, area, and perimeter using that fixed ratio. Geometry and trigonometry students, and tradespeople such as carpenters and framers use 30-60-90 triangles constantly — they arise naturally from bisecting an equilateral triangle and show up in roof framing, stair stringers, and any layout involving a 30° or 60° angle.

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