Question

What is the ratio of the area of a regular hexagon to the area of the hexagon which uses the midpoints of the first hexagon's sides as its vertices? Express your answer as a common fraction.

Answer 1

If you draw a line connecting two midpoints of the hexagon with side 1, then each half is a leg of a 30-60-90 right triangle with hypotenuse 1/2. Si the line joining the midpoints has length 1/2 √3
so the area of the smaller hexagon is 3/4 that of the larger