Find the side of a regular hexagon and the radius of the circumscribed circle if the radius of the inscribed

Find the side of a regular hexagon and the radius of the circumscribed circle if the radius of the inscribed circle is a) 17 cm b) 18√ 3

Let’s construct the radii ОC and ОD of the circumscribed circle.

The formed triangle OCD is equilateral, since in a regular hexagon the length of its edge is equal to the radius of the circumscribed circle.

Then the radius ОН of the inscribed circle is the height, median and bisector of the equilateral triangle ОD.

Then OH = CD * √3 / 2 = 17 cm.

CD = 17 * 2 / √3 = 34 / √3 = 34 * √3 / 3 cm.

Then R = СD = 34 * √3 / 3 cm.

If OH = r = 18 * √3 cm, then R = CD = 18 * √3 * 2 / √3 = 36 cm.

Answer: The side of the hexagon is equal to the radius of the circumscribed circle and is equal to: a) 34 * √3 / 3 cm, b) 36 cm.