Calculate all side lengths of a 45-45-90 isosceles right triangle from any one known side using fixed ratio relationships.
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Enter any one side of a 45-45-90 right triangle to calculate all sides and properties. The ratio is leg : leg : hypotenuse = 1 : 1 : √2.
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Triangle Geometry
A 45-45-90 triangle is a special right triangle with two equal 45-degree angles and one 90-degree angle. Its sides follow a fixed ratio of 1 : 1 : sqrt(2), so knowing any one side determines the other two immediately.
Because the two acute angles are equal, the two legs are also equal. If each leg has length a, the hypotenuse is a * sqrt(2). This relationship comes from the Pythagorean theorem applied to two equal legs: a squared plus a squared equals c squared, so c equals a times the square root of two.
A 45-45-90 triangle is also called an isosceles right triangle because it is both isosceles (two equal sides) and right-angled.
From any one known measurement the remaining sides and the area can be found directly.
leg = hypotenuse / sqrt(2)
Each leg equals the hypotenuse divided by the square root of two.
hypotenuse = leg * sqrt(2)
The hypotenuse is the leg times the square root of two.
Area = leg^2 / 2
The area is half the square of the leg length.
The 45-45-90 triangle appears when cutting a square diagonally, designing mitre joints, or laying out equal-pitch roof valleys. It is also the basis of the unit-circle values for 45 degrees in trigonometry.
Frequently asked questions
7 * sqrt(2), which is approximately 9.90.
A square, because the two equal legs form four right angles at the corners.
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