Two types of special right triangles appear commonly in geometry, the "45-45-90 triangle" and the "30-60-90 triangle." Knowing the ratios of the sides of these special right triangles allows one to quickly calculate various lengths in geometric problems. More interestingly, using these ratios allows one to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, & 60°.

The 45-45-90 Triangle

This is a triangle whose three angles respectively measure 45°, 45°, and 90°. The sides are in the ratio

$1:1:\backslash sqrt\{2\}.$

A simple proof. Suppose you have such a triangle with legs a and b and hypotenuse c. Suppose that a = 1. Since two angles measure 45°, this is an isosceles triangle and we have b = 1. The fact that $c=\backslash sqrt\{2\}$ follows immediately from the Pythagorean Theorem.

The 30-60-90 Triangle

This is a triangle whose three angles respectively measure 30°, 60°, and 90°. The sides are in the ratio

$1:\backslash sqrt\{3\}:2.$

The proof of this fact is obvious using trigonometry. Although the geometric proof is less apparent, it is equally trivial:

Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30-60-90 triangle with hypotenuse of length 2, and base BD of length 1.

The fact that the remaining leg AD has length $\backslash sqrt\{3\}$ follows immediately from the Pythagorean Theorem.

Euclidean plane geometry