45-45-90 Triangle — Definition, Formula & Examples

A 45-45-90 triangle is a right triangle whose two acute angles each measure 45°. Because the two legs are always equal in length, it is also an isosceles right triangle, and its hypotenuse is always

\sqrt{2}

times the length of a leg.

A 45-45-90 triangle is a right isosceles triangle with interior angles of $\frac{\pi}{4}$ , $\frac{\pi}{4}$ , and $\frac{\pi}{2}$ radians. If each leg has length $a$ , the Pythagorean theorem gives the hypotenuse as $a\sqrt{2}$ , yielding the fixed side ratio $1 : 1 : \sqrt{2}$ .

Key Formula

\text{leg} : \text{leg} : \text{hypotenuse} = a : a : a\sqrt{2}

Where:

$a$ = The length of each leg of the triangle
$a\sqrt{2}$ = The length of the hypotenuse

How It Works

You only need one side length to find the other two. If you know a leg, multiply it by

\sqrt{2}

to get the hypotenuse. If you know the hypotenuse, divide it by

\sqrt{2}

(equivalently, multiply by

\frac{\sqrt{2}}{2}

) to get each leg. This ratio holds for every 45-45-90 triangle regardless of size.

Worked Example

Problem: A 45-45-90 triangle has a hypotenuse of length 10. Find the length of each leg.

Set up the ratio: In a 45-45-90 triangle, the hypotenuse equals the leg times √2.

a\sqrt{2} = 10

Solve for the leg: Divide both sides by √2 and rationalize the denominator.

a = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}

Answer: Each leg is

5\sqrt{2} \approx 7.07

units long.

Why It Matters

Designers, architects, and engineers encounter 45-45-90 triangles whenever a square is cut along its diagonal. In trigonometry, this triangle is the source of the exact values

\sin 45° = \cos 45° = \frac{\sqrt{2}}{2}

, which appear repeatedly on standardized tests and in calculus.

Common Mistakes

Mistake: Multiplying the hypotenuse by √2 instead of dividing by √2 when finding a leg.

Correction: The hypotenuse is the longest side. To go from hypotenuse to leg, divide by √2 (or multiply by √2/2). Only multiply by √2 when going from a leg to the hypotenuse.