A 45-45-90 triangle is a right isosceles triangle — both legs equal, and the hypotenuse-to-leg ratio is always exactly the square root of 2.
How it works
Multiplying a known leg by the square root of 2 gives the hypotenuse; dividing a known hypotenuse by the square root of 2 gives the (equal) leg length.
What this does not include
This does not include general right triangles with unequal legs — for those, use this site’s hypotenuse calculator or right-triangle side-and-angle calculator instead.
How to use this calculator
- Choose what you know, then enter the value.
A worked example
A 45-45-90 triangle with a leg of 5: hypotenuse = 5 × √2 = 7.071068.
Given a hypotenuse of 10, each leg is 10 ÷ √2 = 7.071068 — the reverse calculation.
What the variables mean
| Variable | Meaning |
|---|---|
| Leg | Either of the two equal legs |
| Hypotenuse | The longest side, opposite the right angle |
Edge cases worth knowing
Both legs are always equal by definition — a 45-45-90 triangle is just a right isosceles triangle, so knowing one leg gives the other for free.
A leg of zero collapses the triangle entirely, so the calculator declines to show a result — there’s no meaningful shape left to describe.
Frequently asked questions
Why is the hypotenuse-to-leg ratio always the square root of 2?
Because both legs are equal, the Pythagorean theorem simplifies to leg² + leg² = hypotenuse², which reduces to hypotenuse = leg × √2.
Where does a 45-45-90 triangle show up in practice?
Cutting a square diagonally in half, mitered corners at 45 degrees, and many standard architectural and carpentry angles.
Are both non-right angles always exactly 45 degrees?
Yes — since the two legs are equal, the two non-right angles must also be equal, and together with the 90-degree angle they sum to 180, making each exactly 45.
