45-45-90 Triangle Calculator
A 45-45-90 triangle is an isosceles right triangle with both legs equal and a hypotenuse of leg × √2. Enter the leg or the hypotenuse and get all dimensions plus area and perimeter.
Known side
Side opposite the right angle; equals leg × √2
- 1
Leg (both equal, given)
5 - 2
Hypotenuse = leg × √2
5 × √2 = 7.0711
How does this calculator work?
A 45-45-90 (isosceles right) triangle always has sides in ratio 1 : 1 : √2. If each leg is x, the hypotenuse is x√2 ≈ 1.414x and the area is x²/2. Enter either the leg or hypotenuse and get all other dimensions, area, and perimeter instantly.
Formula
How this is calculated
A 45-45-90 triangle has angles 45°, 45°, and 90°. Because the two acute angles are equal, the triangle is isosceles — the two legs (sides forming the right angle) are also equal. Let each leg be x. By the Pythagorean theorem, the hypotenuse c = √(x² + x²) = √(2x²) = x√2. This ratio 1 : 1 : √2 is fixed regardless of the size of the triangle.
Given the leg x, the hypotenuse is x × √2 ≈ 1.4142x. Given the hypotenuse h, each leg is h / √2 = h√2 / 2. The area equals half the product of the two legs: ½ × x × x = x² / 2. The perimeter is 2x + x√2 = x(2 + √2).
This special triangle appears frequently in geometry, construction, and trigonometry. It is produced by cutting a square diagonally, and its angles match the π/4 radian marks on the unit circle — sin 45° = cos 45° = 1/√2 ≈ 0.7071. Results are shown to four decimals; the exact hypotenuse for legs of 1 is √2, which is irrational.
Frequently asked questions
In any triangle, equal angles are opposite equal sides. A 45-45-90 triangle has two 45° angles, so the two sides opposite those angles (the legs) must be equal. Only the hypotenuse, opposite the unique 90° angle, differs.
Divide the hypotenuse by √2 (≈ 1.4142). Equivalently, multiply the hypotenuse by √2/2 ≈ 0.7071. For example, a hypotenuse of 10 gives legs of 10/√2 ≈ 7.071.
Cutting a square along a diagonal creates two 45-45-90 triangles. The diagonal is the hypotenuse and the sides of the square are the legs. This relationship is used in woodworking, construction (mitre cuts), and coordinate geometry.
Also known as
TG we-Calculate Editorial Team. (2026). 45-45-90 Triangle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-45-45-90-calculator
TG we-Calculate Editorial Team. "45-45-90 Triangle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-45-45-90-calculator.
TG we-Calculate Editorial Team, "45-45-90 Triangle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-45-45-90-calculator
@misc{wecalculate_triangle_45_45_90_calculator, title = {45-45-90 Triangle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-45-45-90-calculator}}, year = {2026}, note = {TG we-Calculate} }
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