45-45-90 triangle
Also known as: isosceles right triangle
A 45-45-90 triangle is a right triangle whose two acute angles both measure 45°, making it an isosceles right triangle. Its sides are always in the ratio x : x : x√2, so the hypotenuse equals a leg times √2.
A 45-45-90 triangle is one of the two special right triangles (the other is the 30-60-90). Its angles measure 45°, 45°, and 90°, and because the two acute angles are equal, the two legs opposite them are equal too — it is an isosceles right triangle. You can picture it as a square cut in half along its diagonal.
The side lengths always follow the ratio x : x : x√2 — two equal legs of length x and a hypotenuse of x√2. This comes straight from the Pythagorean theorem: x² + x² = c² gives c = x√2. So if each leg is 5, the hypotenuse is 5√2 (about 7.07). Working backward, if the hypotenuse is 8, each leg is 8/√2, which simplifies to 4√2.
Knowing this ratio means you can find every side of the triangle from any single side — no Pythagorean calculation needed. The 45-45-90 relationship also underlies the exact trigonometric values sin 45° = cos 45° = √2/2, and it appears constantly in problems involving squares, diagonals, and distance.
The GRE, CLT, and ACT all test 45-45-90 triangles regularly, often disguised inside squares or coordinate-geometry problems. Memorizing the x : x : x√2 ratio is one of the highest-value shortcuts in test-prep geometry.
Key takeaways
- A 45-45-90 triangle has angles of 45°, 45°, and 90°, with two equal legs — it is an isosceles right triangle.
- The sides are always in the ratio x : x : x√2.
- Hypotenuse = leg × √2; leg = hypotenuse ÷ √2.
- Cutting a square along its diagonal produces two 45-45-90 triangles.
- The ratio follows directly from the Pythagorean theorem: x² + x² = (x√2)².