Formula and Method for the 45-45-90 Triangle
A 45-45-90 triangle is a right triangle whose two acute angles are both 45°, which makes it isosceles: the two legs opposite those 45° angles are equal in length. Because the angles are fixed, the side lengths always sit in a fixed ratio — leg : leg : hypotenuse = 1 : 1 : √2. Knowing just one side is enough to determine the other two, plus the triangle's area and perimeter.
How the calculation works
If you know a leg (a), the Pythagorean theorem gives the hypotenuse directly: a² + a² = c², so 2a² = c², and c = a√2 ≈ 1.4142 × a. If instead you know the hypotenuse (c), solve the same relationship for the leg: a = c / √2, which is equivalent to a = c × √2⁄2 ≈ 0.7071 × c. Once both legs are known, the area follows from treating the two perpendicular legs as base and height: Area = (a × a) / 2 = a²/2. The perimeter is simply the sum of all three sides: P = a + a + c = 2a + c.
Common mistakes
- Mixing up leg and hypotenuse: the hypotenuse is always the longest side (opposite the 90° angle) and is always longer than either leg — if your "known" side isn't clearly the longest, it's a leg, not the hypotenuse.
- Forgetting to divide by √2: converting a hypotenuse into a leg requires dividing by √2 (≈1.4142), not multiplying — multiplying would make the leg longer than the hypotenuse, which is impossible.
- Squaring only one side for area: the area uses leg² / 2, not leg × hypotenuse / 2 — mixing a leg with the hypotenuse in the area formula overstates the result.
Real-world applications
- Carpentry and framing use 45-45-90 triangles for mitered corners, diagonal bracing, and squaring up right-angle joints.
- A square's diagonal splits it into two 45-45-90 triangles, so this ratio shows up whenever you need a square's diagonal from its side (or vice versa).
- Surveying and CAD drafting use the 1 : 1 : √2 ratio to lay out 45° cuts, ramps, and roof pitches without a protractor.
- Physics problems involving forces or velocities at 45° (e.g., projectile launch angles) resolve into equal perpendicular components, mirroring this triangle's geometry.