45-45-90 Triangle Calculator

A 45-45-90 triangle is a special right triangle with two 45° interior angles and one 90° right angle. Because both acute angles are equal, it is also an isosceles triangle, meaning the two legs are always the same length. You can think of it as a square cut in half diagonally.

The sides follow the ratio 1 : 1 : √2. If each leg has length a, then the hypotenuse equals a√2. This predictable ratio means you never need trigonometric functions — just multiply or divide by √2.

If you know leg a, then leg b = a, hypotenuse c = a√2, area = a²/2, and perimeter = a(2 + √2). All other properties follow directly from this single measurement.

Divide the hypotenuse by √2 (approximately 1.41421). So if c is the hypotenuse, each leg a = c / √2 = c√2 / 2. For example, a hypotenuse of 10 gives legs of about 7.0711 each.

Given perimeter P = a(2 + √2), solve for the leg: a = P / (2 + √2). Then compute the area as a² / 2. For example, with perimeter 10, leg a ≈ 2.9289 and area ≈ 4.289.

It earns both names because it contains a 90° angle (right triangle) and has two equal sides — the two legs (isosceles triangle). The equal legs arise directly from the two equal 45° angles, since in any triangle equal angles face equal sides.

For a 45-45-90 triangle with leg a: the inradius r = a(√2 − 1) / √2 = a(1 − 1/√2), and the circumradius R = c/2 = a√2/2. The circumradius is simply half the hypotenuse, since in any right triangle the hypotenuse is the diameter of the circumscribed circle.

They appear when a square is cut diagonally, in roof pitch calculations, in drafting and design (45° set squares), in trigonometry (sin 45° = cos 45° = √2/2), and in many geometry and engineering problems. Their clean ratio makes mental calculation straightforward.