Question

A regular hexagon and an equilateral triangle have equal perimeters. what is the ratio of the area of the hexagon to the area of the triangle? express your answer as a common fraction.

Answer 1

In this item, we let the perimeter of both polygons be P. The lengths of each side are calculated below.

Hexagon:   s = (p/6) = p/6
Triangle:     s = (p/3) = p/3

The areas of each polygon are also calculated below. It is noted that the polygons are regular (meaning, each side and angle are equal).

Area of Hexagon:    A = 3√3/2 a²
 Substituting the known values,
                     A = 3√3/2(P/6)²
Simplifying,
                      A = √3/24P²

For the triangle,
              A = √3/4a²

Substituting,
                A = √3/4 (P/3)²
Simplifying,
               A = √3/36P²

The ratio is equal to:
              ratio = (√3/24P²) / (√3/36P²)
                ratio = 3/2
                    ratio = 1.5