45-45-90 Triangle Calculator

Isosceles right triangle calculator

Angle α is fixed at 45° (so β is always 45° too, and the two legs are always equal). Enter any one remaining value — a leg, the hypotenuse, the altitude, the area, or the perimeter — pre-loaded here with a 12 in square tile cut corner to corner.

What makes 45-45-90 a "special" right triangle

A 45-45-90 triangle — also called an isosceles right triangle — is a right triangle whose two acute angles are both exactly 45°. Because the angles opposite the two legs are equal, the legs themselves must be equal in length, and the hypotenuse is always the leg length multiplied by √2 (approximately 1.41421356). That single constant is all you need to convert between a leg and the hypotenuse in either direction, with no trigonometry table required.

This calculator pins α (and therefore β) to 45° and lets you supply any one of the six remaining quantities — either leg, the hypotenuse, the altitude to the hypotenuse, the area, or the perimeter — then runs the same general-purpose solver used across this site to fill in everything else.

45-45-90 triangle diagram

Both legs are x; the tick marks show they are equal. The hypotenuse is x√2, so the sides are in the ratio 1 : 1 : √2, and both acute angles are 45°.

<figure>
  <img src="https://righttrianglecalculators.com/images/45-45-90-triangle.png" width="1200" height="800" style="max-width: 100%; height: auto;" alt="A 45-45-90 triangle: two equal legs x and the hypotenuse x√2, with both acute angles 45°.">
  <figcaption>45-45-90 triangle side ratios by <a href="https://righttrianglecalculators.com/45-45-90-triangle-calculator">Right Triangle Calculators</a> is licensed under <a href="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>.</figcaption>
</figure>

Worked example: cutting a 12 in tile corner to corner

Cut a square tile along its diagonal and each half is a 45-45-90 triangle: the tile's edges are the two equal legs, and the cut is the hypotenuse. For a 12 in tile, the other leg is also exactly 12 in, and the cut is c = 12 × √2 ≈ 16.971 in, so check that your tile cutter can cut at least 17 in on the diagonal. Each half covers (1/2) × 12 × 12 = 72 in², exactly half the tile's 144 in².

The rest of the triangle follows: the perimeter is 12 + 12 + 12√2 ≈ 40.971 in, the altitude to the hypotenuse is (12 × 12) / 16.971 ≈ 8.485 in (exactly half the hypotenuse, as it always is in a 45-45-90 triangle), the inradius is (12 + 12 − 12√2)/2 ≈ 3.515 in, and the circumradius is half the hypotenuse, ≈ 8.485 in. The altitude and the circumradius come out equal. That's not a coincidence of this example but a property of every 45-45-90 triangle: with equal legs, both are exactly half the hypotenuse.

When you'd use a 45-45-90 triangle

45-45-90 triangles appear whenever a square or a right-angle corner is cut along its diagonal — mitered picture frames, framing carpentry, cutting square tiles or plywood corner to corner, and countless geometry and trigonometry problems. If your triangle instead has a 30° or 60° angle, use the 30-60-90 triangle calculator, or switch between both types in the special right triangle calculator. For a general right triangle with any angle at all, use the full right triangle calculator.

45-45-90 triangle FAQ

See the full right triangle FAQ for more general questions.

Cite this page

To cite this calculator in schoolwork or a report, you can use this reference (APA style):

Right Triangle Calculators. (2026, October 10). 45-45-90 triangle calculator. https://righttrianglecalculators.com/45-45-90-triangle-calculator

References

The formulas and definitions on this page match these standard references.

1. Weisstein, Eric W. “Isosceles Right Triangle.” MathWorld, Wolfram Research.
2. Weisstein, Eric W. “Pythagoras’s Constant.” MathWorld, Wolfram Research.
3. Weisstein, Eric W. “Pythagorean Theorem.” MathWorld, Wolfram Research.