The 45-45-90 Triangle

Part of the Triangles section of Coddy's Geometry journey. Lesson 45 of 71.

Cut a square along its diagonal. Each half is a right triangle with two equal legs and two 45° angles. A triangle with these angles is called a 45-45-90 triangle. To find its hypotenuse, use the Pythagorean theorem with both legs the same. For legs of 3:

$A right isosceles triangle with both legs 3 and hypotenuse c, the two acute angles marked 45 degrees$

Two legs of 3 and two angles of 45°

c = √(3^2 + 3^2) = √(18) = 3√(2)

The same calculation works for any leg x: √(x^2 + x^2) = √(2x^2) = x √(2). So in a 45-45-90 triangle the hypotenuse is the leg times the root of 2. The root of 2 is about 1.41, so the hypotenuse is about 1.41 times the leg. Legs of 5 give a hypotenuse of 5√(2). Legs of 10 give 10√(2). Once you know the pattern, you only need one multiplication.