Question

Each side of a regular hexagon measures a cms. By joining the mid points of each side, another hexagon is formed inside it. Find the ratio of the areas of outer and the inner hexagons.

Answer 1

For simplicity, let us consider convex & regular polygons only. The ratio of the perimeter of the inner polygon to that of the outer is always 1/2. However, the ratio of the areas is as follows: Triangle — 1/4, Square - 1/2, Pentagon - 0.6545, Hexagon — 0.75. Of course, the two polygons are always similar. For understanding this better, I suggest you read this