45-45-90 Triangle Calculator
From one side of a 45-45-90 right isosceles triangle, get both legs and the hypotenuse in the 1 : 1 : √2 ratio, with area and perimeter.
How it works
leg : leg : hypotenuse = 1 : 1 : √2 · Area = ½ · leg² · Perimeter = 2·leg + hypotenuse
A 45-45-90 triangle is a right isosceles triangle: the two 45° angles make its legs equal, so the sides follow the ratio 1 : 1 : √2. The hypotenuse is always a leg times √2, straight from the Pythagorean theorem on two equal legs. Given the hypotenuse instead, divide by √2 to recover a leg. Since the equal legs are a base and perpendicular height, the area is half the square of a leg.
Worked example
With a leg of 1: the other leg is also 1 and the hypotenuse is 1 × √2 ≈ 1.4142. The area is ½ × 1² = 0.5 and the perimeter is 1 + 1 + 1.4142 ≈ 3.4142. If you enter the hypotenuse as 1.4142 instead, dividing by √2 returns a leg of 1 — the same triangle.
Sources: Special right triangle — Wikipedia
Frequently asked questions
Why are the two legs equal?
The triangle has two 45° angles, and equal angles sit opposite equal sides. That makes it isosceles, so the two legs meeting at the right angle are the same length.
How long is the hypotenuse?
Exactly leg × √2 ≈ 1.4142 × leg. It comes from the Pythagorean theorem: hypotenuse² = leg² + leg² = 2·leg², so the hypotenuse is √2 times a leg.
Where does this triangle show up?
It is the diagonal of a square: cut a unit square corner to corner and each half is a 45-45-90 triangle with legs 1 and hypotenuse √2. That is why it appears constantly in geometry, construction and design.
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This calculator is for educational purposes. Double-check important results before acting on them.