Hello, arenkun!

A circle is circumscribed about a hexagon.

The area outside the hexagon but inside the circle is 15 mē.

(a) Compute the radius of the circle.

(b) Compute the area of the hexagon. Code:

A
* o *
* *
* *
F o o B
* *
* * *
* O *
E o o C
* *
* *
* o *
D

The center of the circle is $\displaystyle O.$

The vertices of the hexagon are: .$\displaystyle A,B,C,D,E,F.$

Draw chords: .$\displaystyle AB, BC, CD, DE, EF, FA$

. . .and radii: .$\displaystyle OA, OB, OC, OD, OE, OF$

Assuming it is a *regular* hexagon, all the chords and radii are equal to the radius $\displaystyle r.$

The area of the circle is: .$\displaystyle \pi r^2$

The hexagon is comprised of six congruent equilateral triangles of side $\displaystyle r.$

The area of one triangle is: .$\displaystyle \frac{\sqrt{3}}{4}r^2$

The area of the hexagon is: .$\displaystyle 6 \times\frac{\sqrt{3}}{4}r^2 \:=\:\frac{3\sqrt{3}}{2}\,r^2$

The difference of the areas is 15 mē: . $\displaystyle \pi r^2 - \frac{3\sqrt{3}}{2}r^2 \:=\:15$

. . $\displaystyle \left(\pi - \frac{3\sqrt{3}}{2}\right)r^2 \:=\:15 \quad\Rightarrow\quad \left(\frac{2\pi - 3\sqrt{3}}{2}\right)r^2 \:=\:15

$

Hence: .$\displaystyle r^2 \:=\:\frac{30}{2\pi-3\sqrt{3}} \quad\Rightarrow\quad r \;=\;\sqrt{\frac{30}{2\pi-3\sqrt{3}}}$

Therefore: .$\displaystyle r \;=\;5.253385683\text{ m} \;\;(a)$

The area of the hexagon is: . $\displaystyle \frac{3\sqrt{3}}{2}\,r^2 \;=\;\frac{3\sqrt{3}}{2}\left(\frac{30}{2\pi-3\sqrt{3}}\right) \;=\;71.70186611\text{ m}^2\;\;(b)$