# What is a 30-60-90 triangle? Please give an example.

Nov 13, 2015

A 30-60-90 triangle is a right triangle with angles ${30}^{\circ}$, ${60}^{\circ}$, and ${90}^{\circ}$ and which has the useful property of having easily calculable side lengths without use of trigonometric functions.

#### Explanation:

A 30-60-90 triangle is a special right triangle, so named for the measure of its angles. Its side lengths may be derived in the following manner.

Begin with an equilateral triangle of side length $x$ and bisect it into two equal right triangles. As the base is bisected into two equal line segments, and each angle of an equilateral triangle is ${60}^{\circ}$, we end up with the following

Because the sum of the angles of a triangle is ${180}^{\circ}$ we know that $a = {180}^{\circ} - {90}^{\circ} - {60}^{\circ} = {30}^{\circ}$

Furthermore, by the Pythagorean theorem, we know that
${\left(\frac{x}{2}\right)}^{2} + {h}^{2} = {x}^{2}$
$\implies {h}^{2} = \frac{3}{4} {x}^{2}$
$\implies h = \frac{\sqrt{3}}{2} x$

Therefore a 30-60-90 triangle with hypotenuse $x$ will look like

For example, if $x = 2$, the side lengths of the triangle will be $1$, $2$, and $\sqrt{3}$