45-45-90 Triangle Calculator

Background.

How far is it from corner to corner across a square? Draw the diagonal and you have cut the square into two 45-45-90 triangles, each with the square's sides as its legs and the diagonal as its hypotenuse. Pythagoras gives that diagonal as √(L² + L²) = L√2, so a square with 10 m sides measures 14.1421 m corner to corner, and the same factor of about 1.4142 applies to a floor tile, a sheet of plywood or a square window frame.

That is the whole of the triangle's geometry. Its two legs are equal, its two acute angles are both 45°, and its sides stand in the ratio 1 : 1 : √2 at every size. Because nothing about the shape can vary except its scale, one measurement settles everything: a leg, the hypotenuse, the area or the perimeter. Going from a leg to the hypotenuse multiplies by √2; coming back divides by it, which is the same as multiplying by √2 ÷ 2, about 0.7071.

The triangle's other name, isosceles right triangle, describes it equally well, and it turns up wherever square corners do. A mitred picture frame joins two 45° cuts, a diagonal brace across a square opening is a hypotenuse, and a roof pitched at 12 in 12 rises at exactly 45°. In trigonometry it supplies sin 45° = cos 45° = √2 ÷ 2 and tan 45° = 1.

One quantity behaves unlike the others. The area, half the leg squared, stays a tidy number whenever the leg is one: a leg of 10 gives an area of exactly 50, even though the hypotenuse beside it has a decimal expansion that never ends.

What is 45-45-90 triangle calculator?

A 45-45-90 triangle is a right triangle whose two other angles are both 45°, which forces its two legs to be equal. Wolfram MathWorld lists it as the isosceles right triangle and gives, for legs of length a, a hypotenuse of √2·a and an area of ½a². It is exactly half of a square cut along a diagonal, a construction that needs no trigonometry: Euclid builds the square in Elements Book I, Proposition 46, and Proposition 47, the Pythagorean theorem, supplies the diagonal.

How to use this calculator.

1. 01Pick the measurement you have: leg, hypotenuse, area or perimeter. There is only one leg option because the two legs are the same length.
2. 02To find the diagonal of a square, enter the square's side as the leg and read the hypotenuse. To size a square from a measured diagonal, enter the diagonal as the hypotenuse and read the leg.
3. 03Type the value in any unit you like; the lengths that come back share it and the area is in that unit squared.
4. 04Sanity-check the area in leg mode: it should be exactly half the leg squared. A long decimal for a whole-number leg suggests you entered the hypotenuse by mistake.

The formula.

Every mode reduces the input to the leg L and builds the triangle from it. Hypotenuse = L√2. Area = L² ÷ 2, since the legs meet at the right angle and serve as base and height. Perimeter = L + L + L√2 = L(2 + √2), about 3.4142 L. The altitude from the right angle to the hypotenuse is L × L ÷ (L√2) = L ÷ √2, which is exactly half the hypotenuse. The inradius, the radius of the largest circle that fits inside, follows from the right-triangle rule r = (a + b − c) ÷ 2, giving L(2 − √2) ÷ 2, about 0.2929 L.

Working backwards, a hypotenuse h gives L = h ÷ √2, an area A gives L = √(2A), and a perimeter P gives L = P ÷ (2 + √2). Substituting L = h ÷ √2 into the area shows A = h² ÷ 4, so the area is a quarter of the square of the hypotenuse.

The altitude has a second meaning. The midpoint of a right triangle's hypotenuse is the centre of its circumscribed circle, so the circumradius is always half the hypotenuse; in this triangle, and only because the legs are equal, the altitude lands on the same point and has the same length.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — 'Isosceles Right Triangle' (live revision retrieved 2026-07-29). States the angles of 45°, 45° and 90°, that for legs of length a 'the hypotenuse has length √2 a', and the area ½a². Open access.
2. [2]Euclid, Elements, Book I, Proposition 46 — 'To describe a square on a given straight line.' The parent figure of the diagonal-bisection derivation. David E. Joyce web edition of the Heath translation, Clark University; statement confirmed verbatim on retrieval 2026-07-29. Open access.
3. [3]Euclid, Elements, Book I, Proposition 47 — 'In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.' Applied to the halved square it gives the hypotenuse as √(2L²) = L√2, an entirely geometric derivation that uses no trigonometry and serves as the independent check on this page's constant. Joyce web edition; confirmed verbatim on retrieval 2026-07-29. Open access.
4. [4]Wolfram Research, MathWorld — 'Pythagoras's Constant' (live revision retrieved 2026-07-29). Gives the expansion √2 = 1.4142135623… and states that the irrationality of √2 'means that it cannot be expressed as a ratio p/q of integers p and q' — the basis for this page's scope limit that no 45-45-90 triangle has all-rational sides. Open access.