The 30-60-90 Triangle

Part of the Triangles section of Coddy's Geometry journey. Lesson 47 of 71.

Cut an equilateral triangle in half along its height. Each half is a right triangle with angles 30°, 60° and 90°. A triangle with these angles is called a 30-60-90 triangle. Its hypotenuse is a full side of the equilateral triangle, and its short leg is half a side. So the hypotenuse is twice the short leg.

$An equilateral triangle with sides 2x cut in half by its dashed height, one half having angles 30, 60 and 90 degrees and a short leg of x$

Half an equilateral triangle of side 2x: the short leg is x and the hypotenuse is 2x

The long leg is the height of the equilateral triangle. The Pythagorean theorem gives its length. With short leg x and hypotenuse 2x, the long leg is √((2x)^2 - x^2) = √(3x^2) = x √(3). So the three sides are x, x √(3) and 2x. The short leg faces the 30° angle, and the long leg faces the 60° angle.

$A right triangle with angles 30 and 60 degrees, its short leg x, its long leg x times the root of 3 and its hypotenuse 2x$