Enter any one known value — leg (a), hypotenuse (c), area, or perimeter — and the 45-45-90 Triangle Calculator solves the rest. You'll get both legs, the hypotenuse, height, area, perimeter, inradius, and circumradius for your special right triangle. Also try the General Triangle Solver Calculator.
Hypotenuse (c)
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Leg a
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Leg b
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Height (h)
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Area
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Perimeter
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Inradius (r)
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Circumradius (R)
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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. See also our calculate Volume (V), Radius (r) & Diameter (D) — Hemisphere.
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. You might also find our Area Calculator useful.
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.