Question

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

Answer 1

"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=sqrt3/2a`

Again its area `A=1/2*a*sqrt3/2*a=sqrt3/4a^2`

Now we have `a=(2h)/sqrt3`

So expressing A interms of h

`A=sqrt3/4*("2h"/sqrt3)^2=1/3h^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 `A/2=1/6h^2`