Question

The difference in the areas of the regular hexagon circumscribing a circle of radius 10 cm and the regular hexagon inscribed in the circle is ______.

Answer 1

The difference in the areas of the regular hexagon circumscribing a circle of radius 10 cm and the regular hexagon inscribed in the circle is `underlinebb(50sqrt(3)  cm^2)`.

Explanation:

Let r = 10 cm.

For the circumscribing hexagon the circle is the hexagon's in‑circle (apothem = r).

Side `a_1 = (2r)/sqrt(3)` 

So area = `1/2 (6a_1)r` 

= 3a₁r

= `2sqrt(3) r^2` 

= `200sqrt(3)  cm^2`

For the inscribed hexagon the circle is the circumcircle (side a₂ = r).

Area = `((3sqrt(3))/2)a_2^2` 

= `150sqrt(3)  cm^2`

Difference = `200sqrt(3) - 150sqrt(3)`

= `50sqrt(3)  cm^2`