Question

The ratio of the areas of a square and a regular hexagon with inscribed in a circle is

Answer 1

Let the radius of the circle = r

Then the diameter of the circle = the diagonal of the square = 2r

Let a side of the square = x

Then 2x^2 = (2r)^2 = 4r^2

The area of the square = x^2 = 2r^2

The hexagon can be partitioned into 6 equilateral triangles whose sides = r

If the sides of an equilateral triangle = r then its area = r^2 (sqrt 3)/4

So the area of the hexagon = 6 r^2 (sqrt 3)/4 = 3 r^2 (sqrt 3)/2

Thus the ratio of the area of the square to that of the hexagon =

2r^2/[3 r^2 (sqrt 3)/2]

= 4/(3 sqrt 3)

= approx. 0.7698