Prove that the area S of a regular hexagon circumscribed about a circle of radius r can be

Prove that the area S of a regular hexagon circumscribed about a circle of radius r can be found by the formula S = 2√3r ^ 2

Since the hexagon is regular, its large diagonals divide it into six equilateral, equal-sized triangles.

Then the area of the hexagon is: S = 6 * Saov.

The radius r of the inscribed circle is the height of the equilateral triangle AOB, OH = r.

The inner angles of triangle ABC are 60, then Sin60 = OH / OA.

OA = OH / Sin60 = r / (√3 / 2) = 2 * r / √3 cm.

Then Savs = AB * OH / 2 = (2 * r / √3) * r / 2 = r ^ 2 / √3 cm2.

Then S = 6 * r ^ 2 / √3 = 6 * r2 * √3 / √3 * √3 = 2 * r ^ 2 * √3, as required.