Special Right Triangles Calculator
Choose the triangle type and enter one side to find the others from the fixed ratios.
Special right triangles have fixed side ratios. A 45-45-90 triangle has legs equal and a hypotenuse of leg × √2. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the short leg is the base, the long leg is short × √3, and the hypotenuse is short × 2.
How to Use This Calculator
Choose which special triangle you have. For a 45-45-90 triangle, enter either leg — the two legs are equal. For a 30-60-90 triangle, enter the short leg, the side opposite the 30-degree angle, which is the base value the other sides are built from.
The calculator returns all three sides and shows how each came from the ratio. These two triangles appear constantly in geometry and trigonometry because their exact side ratios never change.
How the Calculation Works
A 45-45-90 triangle is half of a square cut along its diagonal, so its two legs are equal and the hypotenuse — the diagonal — is a leg times √2. That single ratio solves the whole triangle from one leg.
A 30-60-90 triangle is half of an equilateral triangle. Its sides are in the ratio 1 : √3 : 2, where the shortest side faces the 30-degree angle, the medium side (short × √3) faces the 60-degree angle, and the hypotenuse (short × 2) faces the right angle.
45-45-90 → leg, leg, leg√2 · 30-60-90 → short, short√3, 2·short
Worked Example
For a 45-45-90 triangle with a leg of 5, the other leg is also 5 and the hypotenuse is 5√2 ≈ 7.07.
For a 30-60-90 triangle with a short leg of 5, the long leg is 5√3 ≈ 8.66 and the hypotenuse is 5 × 2 = 10.
Assumptions and Limitations
- Covers the two standard special right triangles only
- For 30-60-90, enter the short leg (opposite the 30° angle)
- Exact answers involve √2 or √3; decimals shown are rounded
- For other triangles, use the Pythagorean theorem or trigonometry
Who Uses a Special Right Triangles Calculator
Geometry and trigonometry students, along with carpenters and drafters, use this to solve 45-45-90 and 30-60-90 triangles from a single side. It is perfect for homework, roof and stair angles, and any layout that relies on these fixed ratios. Enter one length and get the other sides instantly, with the ratio shown.
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Open calculator →Frequently Asked Questions
What are the special right triangles?
The 45-45-90 and 30-60-90 triangles. They have fixed side ratios — 1 : 1 : √2 and 1 : √3 : 2 — so knowing one side determines the others exactly, without trigonometry.
What is the ratio of a 30-60-90 triangle?
Its sides are in the ratio 1 : √3 : 2. The short leg faces the 30° angle, the long leg (short × √3) faces the 60° angle, and the hypotenuse (short × 2) faces the right angle.
What is the hypotenuse of a 45-45-90 triangle?
It is a leg multiplied by √2. Since both legs are equal, a 45-45-90 triangle with legs of 5 has a hypotenuse of 5√2, about 7.07.
Which side do I enter for a 30-60-90 triangle?
Enter the short leg — the side opposite the 30-degree angle. The calculator builds the long leg and hypotenuse from it using the √3 and 2 multipliers.
Why are these triangles special?
Because their angles force exact, unchanging side ratios, they let you find all sides from one measurement without a calculator's trig functions, which is why they appear so often in geometry and standardized tests.
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