# Given a 30-60-90 triangle in a polygon, for example, how can the apothem be used to find the area of a triangle?

Jun 28, 2016

"Given a 30-60-90 triangle in a polygon" means the polygon is a hexagon.Because in a hexagon the apothem i.e.the perpendicular drawn from its center to any of its sides bisects the equilateral triangle in two equal halves forming two 30-60-90 triangle.So the apothem or the height of the equilateral triangle is known.We are to find out its area.

We know if the side of an equilateral triangle is a then its height $h = \frac{\sqrt{3}}{2} a$

Again its area $A = \frac{1}{2} \cdot a \cdot \frac{\sqrt{3}}{2} \cdot a = \frac{\sqrt{3}}{4} {a}^{2}$

Now we have $a = \frac{2 h}{\sqrt{3}}$

So expressing A interms of h

$A = \frac{\sqrt{3}}{4} \cdot {\left(\frac{\text{2h}}{\sqrt{3}}\right)}^{2} = \frac{1}{3} {h}^{2}$

Now this equation can be used to calculate the area of the equilateral triangle knowing the apothem and the area of 30-60-90 triangle will be $\frac{A}{2} = \frac{1}{6} {h}^{2}$