Right Triangle Calculator · Practical guide
45 45 90 Triangle Hypotenuse
Both legs are equal, and the hypotenuse is always the leg times √2 — about 1.4142 times longer.
At a glance
The Rule
In a 45-45-90 triangle the two legs are equal and the hypotenuse equals leg × √2 ≈ leg × 1.4142. A leg of 10 gives a hypotenuse of 14.1421; a leg of 5 gives 7.0711. Going the other direction, divide the hypotenuse by √2 to recover the leg: a hypotenuse of 20 comes from legs of 20 ÷ 1.4142 ≈ 14.1421.
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Right Triangle Calculator
Enter your own values, then compare the result with the examples below.
Right Triangle Solution
The 1 : 1 : √2 Ratio
The ratio follows directly from the Pythagorean theorem. With both legs equal to x, c = √(x² + x²) = √(2x²) = x√2. Because the two acute angles are equal (45° each), the sides opposite them must be equal too — that is what makes this the isosceles right triangle.
Leg-to-Hypotenuse Lookup
| Leg | Hypotenuse (leg × √2) | Area |
|---|---|---|
| 1 | 1.4142 | 0.5 |
| 2 | 2.8284 | 2 |
| 5 | 7.0711 | 12.5 |
| 10 | 14.1421 | 50 |
| 12 | 16.9706 | 72 |
| 25 | 35.3553 | 312.5 |
A Worked Example
Take a 45-45-90 triangle with legs of 10. The hypotenuse is 10√2 ≈ 14.1421, the area is (10 × 10) ÷ 2 = 50, and the perimeter is 10 + 10 + 14.1421 ≈ 34.1421. This triangle shows up everywhere: cutting a square along its diagonal produces two of them, which is why a square with 10-inch sides has a diagonal of about 14.14 inches. Enter any single side in the calculator above with a 45° angle to get the full solution.
Frequently Asked Questions
How is a 45-45-90 triangle related to a square?
Slicing a square along its diagonal creates two 45-45-90 triangles. The square's diagonal is exactly side × √2, which is the same relationship as leg to hypotenuse.
What is the hypotenuse of a 45-45-90 triangle with leg 1?
It is √2 ≈ 1.4142. This is the classic proof that √2 is irrational — the diagonal of a unit square cannot be expressed as a fraction.
Can a 45-45-90 triangle have whole-number sides?
No. Because the hypotenuse is the leg times the irrational number √2, at least one side must be irrational. If the legs are integers, the hypotenuse never is.
How do 45-45-90 and 30-60-90 triangles differ?
A 45-45-90 has ratio 1:1:√2 and two equal legs. A 30-60-90 has ratio 1:√3:2, where the hypotenuse is twice the short leg and the long leg is the short leg times √3 ≈ 1.732.