Special Right Triangle Calculator

A special right triangle has angles of 45°-45°-90° or 30°-60°-90°, and its sides always come in a fixed ratio: 1 : 1 : √2 for the first and 1 : √3 : 2 for the second. Knowing one side is enough to find the other two exactly. This 30-60-90 and 45-45-90 triangle calculator does it and shows the working.

How to use the special right triangle calculator

1. Pick the triangle. If the two legs are equal or the angles are 45°, it is a 45-45-90 triangle. If the angles are 30° and 60°, or the hypotenuse is twice the shortest side, it is a 30-60-90 triangle.
2. Type the one side you know in its box. In a 30-60-90 triangle the short leg is the one opposite the 30° angle; the long leg is opposite 60°.
3. Read the exact sides. The result line shows all three sides in exact form, the tiles give each one as a decimal too, and the steps show how each was found, including rationalizing when you start from a long leg or a hypotenuse.

You can type exact values. Entering 4√3 as the long leg of a 30-60-90 triangle gives a short leg of exactly 4.

Where the two ratios come from

Neither ratio needs to be memorized blindly. Each triangle is half of a familiar shape, and the Pythagorean theorem gives the missing side.

• 45-45-90. Cut a square with side 1 along its diagonal. The legs are 1 and 1, so the hypotenuse is √1² + 1² = √2. Full details: 45-45-90 triangle.
• 30-60-90. Cut an equilateral triangle with side 2 down the middle. The hypotenuse is 2, the short leg is half the base, 1, and the long leg is √2² − 1² = √3. Full details: 30-60-90 triangle.

Scaling the triangle by any factor x scales every side by x, which gives the general patterns x : x : x√2 and x : x√3 : 2x.

Formulas for every known side

A good habit for 30-60-90 problems is to find the short leg first, whatever you are given. Everything else is a multiple of it.

Worked examples

Area and perimeter

Because the sides are fixed multiples of one length x, area and perimeter have one-line formulas. The calculator uses them and keeps the answers exact.

• 45-45-90 with legs x: area x² over 2, perimeter (2 + √2) x.
• 30-60-90 with short leg x: area √3 over 2x², perimeter (3 + √3) x.

For the default example, a 30-60-90 triangle with short leg 5, the area is 25√3 over 2 ≈ 21.6506 and the perimeter is 15 + 5√3 ≈ 23.6603.

Spotting special triangles in problems

Problems rarely say "this is a 30-60-90 triangle". Look for these shapes:

• The diagonal of a square is always side × √2. A square with 6 m sides has a diagonal of 6√2 ≈ 8.49 m. For rectangles in general, use the rectangle diagonal calculator.
• The height of an equilateral triangle is side × √3 over 2. A triangle with 10 cm sides is 5√3 ≈ 8.66 cm tall. See height of a triangle.
• Regular hexagons split into six equilateral triangles, so their measurements are full of 30-60-90 triangles. A hexagon with 4 cm sides is 4√3 ≈ 6.93 cm across its flat sides.
• Any right triangle with a 30°, 45° or 60° angle, such as a ramp, a ladder against a wall or a roof line.

When the angle is anything else, use trigonometry instead; the guide to Pythagorean theorem or trigonometry explains how to choose, and the right triangle solver handles any angle.

Common mistakes

• Putting √3 on the hypotenuse. In a 30-60-90 triangle the hypotenuse is exactly twice the short leg. The √3 belongs to the long leg.
• Mixing up the legs. The short leg is opposite the smallest angle, 30°. The long leg is opposite 60°.
• Leaving a root in the denominator. 10 over √2 is correct but most teachers want 5√2. The square root simplifier covers rationalizing.
• Using the ratios on the wrong triangle. They only apply when the angles really are 30-60-90 or 45-45-90. Check the angles, or check the sides with a² + b² = c².