30-60-90 triangle calculator, every side from one.

A 30-60-90 triangle calculator finds all three sides of a right triangle whose other angles are 30 and 60 degrees. Its sides are always in the ratio 1 : √3 : 2. Type the short leg, the long leg, the hypotenuse, the area or the perimeter, and this calculator gives every side in exact form (such as 5√3) and as decimals, with the area, the perimeter, a drawing to scale and the working.

How to use the 30-60-90 triangle calculator

1. Under What you know, choose the side you have. The short leg is opposite the 30 degree angle and the long leg is opposite the 60 degree angle.
2. Type its value in the box below. Whole numbers, decimals and fractions such as 7/2 work.
3. Choose a Unit and the Decimal places if you want them.
4. Read the three side cards. The exact form is under each decimal. The working shows the steps and a Pythagoras check.
5. Press Copy result or Copy link to keep the answer.

Why the ratio is 1 : √3 : 2

Take an equilateral triangle with side 2x and cut it in half down the middle. You get two 30-60-90 triangles. The hypotenuse is the original side, 2x. The short leg is half of the base, x. The long leg is the height, and by Pythagoras it is √((2x)² − x²) = √(3x²) = x√3. So the sides are x, x√3 and 2x.

Which side do you have

Short leg s. The long leg is s√3 and the hypotenuse is 2s. For s = 5, the long leg is 5√3 = 8.6603 and the hypotenuse is 10. The area is ½ × 5 × 8.6603 = 21.6506, and the perimeter is 5 + 8.6603 + 10 = 23.6603.

Long leg L. The short leg is L ÷ √3, written exactly as L√3 ÷ 3, and the hypotenuse is 2L ÷ √3. For L = 5, the short leg is 5√3/3 = 2.8868 and the hypotenuse is 10√3/3 = 5.7735.

Hypotenuse H. The short leg is H ÷ 2 and the long leg is (H ÷ 2)√3. For H = 10, they are 5 and 5√3.

Sides for short legs 1 to 5

Find the height of an equilateral triangle

An equilateral triangle with side 10 splits into two 30-60-90 triangles with hypotenuse 10. Its height is the long leg: 5√3 = 8.6603. Its area is two copies of 21.6506, which is 43.3013. This is the quickest way to the height of any equilateral triangle: half the side times √3.

Common problems

• You are not sure which leg is which. The short leg faces the smallest angle, 30 degrees. In the drawing it is the vertical side on the left.
• The area or perimeter result has no exact form. Those are solved with decimals, so only the decimal values are given.
• The angles are 45 and 45. Use the 45-45-90 triangle calculator. For any other right triangle, use the triangle calculator.

For other shapes' areas, see the area and volume calculator.