Determine the maximum ratio of the cross-sectional area of a hexagon vs. the area for the circle that contains such hexagon.

the ratio of the area of the hexagon vs area of the circle that contains it is 0.8269

Explanation:

The area of a hexagon Ah expressed in terms of the distance from the centre to one of the vertex R is

Ah = 6 * (area triangle with angle 2π/6) = 6 * (1/2) * R*R sin (2π/6) = 3*R²*√3/2 =

(3/2) √3*R²

the area of the circle Ac that contains such hexagon is

Ac = π*R²

the ratio of the area of the hexagon vs area of the circle is

r = Ah/Ac = (3/2) √3*R² / (π*R²) = 3*√3 / (2*π) = 0.8269