How a 45-45-90 triangle is solved
The two legs are equal and the hypotenuse is always √2 times a leg, so one measurement fixes the whole triangle.
a : a : c = 1 : 1 : √2 · A = a² ÷ 2 · P = a(2 + √2)
- From a leg a: the hypotenuse is c = a × √2.
- From the hypotenuse: a = c ÷ √2, which is the same as c × √2 ÷ 2.
- From the area: a = √(2 × A). From the perimeter: a = P ÷ (2 + √2).
- The height to the hypotenuse is a ÷ √2, the inradius is a(2 − √2) ÷ 2, and the circumradius is c ÷ 2.
Worked example — Leg 5 cm: c = 5 × √2 = 5√2 ≈ 7.0711 cm. Area = 5² ÷ 2 = 12.5 cm². Perimeter = 5 + 5 + 7.0711 = 17.0711 cm.
Common 45-45-90 triangles
| Leg | Hypotenuse | Area | Perimeter |
|---|---|---|---|
| 1 cm | 1.4142 cm (√2) | 0.5 cm² | 3.4142 cm |
| 2 cm | 2.8284 cm (2√2) | 2 cm² | 6.8284 cm |
| 3 cm | 4.2426 cm (3√2) | 4.5 cm² | 10.2426 cm |
| 5 cm | 7.0711 cm (5√2) | 12.5 cm² | 17.0711 cm |
| 10 cm | 14.1421 cm (10√2) | 50 cm² | 34.1421 cm |
| 12 cm | 16.9706 cm (12√2) | 72 cm² | 40.9706 cm |
Swipe the table sideways to see the rest.
From the leg, in centimetres. Each row is worked out by this calculator.
Things to consider
- Keep √2 in the answer for exact work, and round only at the end.
- The diagonal of a square is its side × √2, which is this triangle's hypotenuse.
- The circumcentre sits at the midpoint of the hypotenuse.
- To check a corner is square, the diagonal of equal sides should be about 1.414 times a side.
Frequently asked questions
What is a 45-45-90 triangle?
A right triangle with two 45° angles. Its two legs are equal, so it is also called a right isosceles triangle.
What is the side ratio of a 45-45-90 triangle?
1 : 1 : √2. The legs are equal and the hypotenuse is √2, about 1.4142, times a leg.
How do I find the hypotenuse from a leg?
Multiply the leg by √2. A 5 cm leg gives 5√2 ≈ 7.0711 cm.
How do I find a leg from the hypotenuse?
Divide by √2, or multiply by √2 ÷ 2. A hypotenuse of 10 gives a leg of 5√2 ≈ 7.0711.
What is the area of a 45-45-90 triangle?
Area = a² ÷ 2, half of a square with side a. A leg of 6 gives 18.
Where does a 45-45-90 triangle appear?
Cut a square along its diagonal and you get two of them. To find the diagonal of a square or any rectangle, use the Rectangle Calculator: Area and Perimeter.
Results are estimates for general information only, not financial, medical or professional advice.