text: my question was some what big so i posted in detail section.? a hexagon(h1) with all sides of length ‘a’, if  midpoints of the sides of hexagon are joined we can get another hexagon(h2), what is the ratio of areas of h2 and h2? h1's area = 6 * area of equilateral triangle with side=1. h2's area = 6 * area of equilateral triangle with side=altitude of h1's equilateral triangle. altitude of h1's equilateral triangle = sqrt(3)/2. thus area ratio h1/h2 = 1^2/(sqrt(3)/2)^2 = 1/.75 = 4/3.