About the Special Right Triangles Calculator
The 30-60-90 and 45-45-90 triangles are the two “special” right triangles whose sides always follow fixed ratios: 1 : √3 : 2 and 1 : 1 : √2. Because the ratios never change, knowing just one side is enough to find the others. Choose the triangle type, say which side you know, and enter its length; this calculator returns every side, the area, the perimeter and the altitude to the hypotenuse.
It is aimed at students learning trigonometry and preparing for exams such as the SAT, ACT and GCSE, where these triangles appear constantly, and at builders and designers working with 45° mitres, 30° roof pitches, hexagons and equilateral shapes.
A 45-45-90 triangle is half of a square cut along its diagonal; a 30-60-90 triangle is half of an equilateral triangle. Those two facts are all you need to derive the side ratios yourself.
How to use the special right triangles calculator
- 1Pick 30-60-90 or 45-45-90.
- 2Choose which side you already know.
- 3Enter its length.
- 4Read all three sides, the area, perimeter and altitude.
Formula and method
A 30-60-90 triangle is half of an equilateral triangle of side 2x, so the hypotenuse is 2x, the short leg (opposite 30°) is x, and by Pythagoras the long leg (opposite 60°) is √(4x² − x²) = x√3. From any known side, first find x — divide the long leg by √3 or the hypotenuse by 2 — then multiply out.
A 45-45-90 triangle is half of a square with side x, so both legs are x and the hypotenuse is the square’s diagonal, x√2. Divide a known hypotenuse by √2 to get the legs. The area is half the product of the legs, and the altitude to the hypotenuse is leg × leg ÷ hypotenuse.
- x
- Short leg (30-60-90) or leg (45-45-90)
- x√3
- Long leg of a 30-60-90 triangle
- 2x, x√2
- Hypotenuse of each triangle type
Worked examples
30-60-90 with short leg 5
With x = 5 the long leg is 5√3 ≈ 8.66 and the hypotenuse is 2 × 5 = 10. The area is ½ × 5 × 8.66 ≈ 21.65 and the perimeter about 23.66.
45-45-90 with legs of 7
Both legs are 7, so the hypotenuse is 7√2 ≈ 9.90. The area is ½ × 7 × 7 = 24.5 — half of a 7 × 7 square.
30-60-90 with hypotenuse 12
Halving the hypotenuse gives the short leg, 6, and the long leg is 6√3 ≈ 10.39. The area is ½ × 6 × 10.39 ≈ 31.18.
Frequently asked questions
What are the side ratios of a 30-60-90 triangle?+
The sides are in the ratio 1 : √3 : 2 — short leg x, long leg x√3 and hypotenuse 2x. The short leg is opposite 30° and the long leg opposite 60°.
What are the side ratios of a 45-45-90 triangle?+
Both legs are equal and the hypotenuse is √2 times a leg, giving 1 : 1 : √2. A leg of 7 means a hypotenuse of 7√2 ≈ 9.90.
How do I find the legs of a 45-45-90 triangle from the hypotenuse?+
Divide the hypotenuse by √2 (or multiply by √2/2 ≈ 0.7071). A hypotenuse of 10 gives legs of about 7.071 each.
How do I find the short leg from the long leg in a 30-60-90 triangle?+
Divide the long leg by √3 (≈ 1.732). A long leg of 9 gives a short leg of 9/√3 = 3√3 ≈ 5.196 and a hypotenuse of about 10.39.
Why are they called special right triangles?+
Their angles give exact, memorable side ratios and exact trig values such as sin 30° = 1/2 and cos 45° = √2/2, so problems can be solved without a calculator.