Given a 30-60-90 triangle in a polygon, for example, how can the apothem be used to find the area of a triangle?
"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
Again its area
Now we have
So expressing A interms of h
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