The formula for the area of a circumscribed hexagon = 1/2 * 6 * a * r where a = side length and r = radius of the circle ( = 1/2 the diameter) so in terms of the diameter and side length its 1/2 * 1/2 * 6 *d * a = 1.5 ad.
You can calculate the value of a by using trigonometry You consider one of the 6 isosceles triangles that make up the hexagon. The vertex angle of these triangles = 60 degrees so:-
tan 30 = 1/2 a / r = 1/2 a / 1/2 d = a/d
a = d tan 30 = sqrt3 d / 3
Substituting for a :-
So the required formula for the area of the hexagon = 1.5 ad
= 1.5 * sqrt3 d * d / 3 = 1/2 sqrt3 d^2 Answer