A regular hexagon can be broken into 6 equal triangles, each one with 3 equal sides and 3 equal angles (thus the angles are all 180/3=60).
In this case the sides of each of these triangles are r, the radius of the circle. Draw it, two of the sides of each triangle are from the origin to the circle's perimeter. That's r. Since the triangles are equilateral, all sides are the same. But that doesn't even matter.
When given the measure of two sides of a triangle, and one angle, the area is:
A = (ab)*sin(c)/2 where a and b are the known sides (in this case each is r) and c is the known angle (in this case 60 degrees).
For each triangle A=r*r*sin(60)/2.
For the whole hexagon it is A(hexagon) = 6*(r*r*sin(60)/2) because there are six such triangles in the hexagon.
That's 93.53 = 3r^2*sin(60). The problem told us A(hexagon) is 93.53.
Sin(60)=.866 (approx.) so this becomes 93.53 = 3r^2*.866 = 2.5980r^2.
Divide both sides by 2.5980
36=r^2
The area of a circle is pi*r^2 = 3.14159*36= (appox) 113.10.
The area of the circle not covered by the hexagon is 113.10 - 93.53= (appox) 19.57 cm^2.
I provide online tutoring ($30/hr) and personal problem solving
($3.50-$5.50 per problem. Contact me (check the profile "website" which is
my email address.
(