A regular hexagon is inscribed in a circle and a regular hexagon is described around. Find: the ratio of their areas.

Let the radius of the circle be R cm.

Then the length of the side of the hexagon inscribed in it is also equal to R, see AB = R.

Determine the area of the inscribed hexagon.

S1 = 3 * √3 * R ^ 2/2 cm2.

The radius of the inscribed circle in a regular hexagon is: R = a * √3 / 2, where a is the side of the described hexagon.

a = 2 * R / √3.

Then the area of the second hexagon is equal to:

S2 = 3 * √3 * (2 * R / √3) ^ 2/2 = 4 * √3 * R ^ 2/2 cm2.

S2 / S1 = (4 * √3 * R ^ 2/2) / (3 * √3 * R ^ 2/2) = 4/3.

Answer: The area ratio is 4/3.