SOLUTION: A circle is inscribed inside a regular hexagon. Find the ratio of the area of the circle to the area of the hexagon.

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Question 1103080: A circle is inscribed inside a regular hexagon. Find the ratio of the area of the circle to the area of the hexagon.
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52870) About Me  (Show Source):
You can put this solution on YOUR website!
.
The area  S%5Bn%5D of a regular  n-sided polygon circumscribed about a circle of the radius R is equal to


S%5Bn%5D = n.R%5E2tan%28pi%2Fn%29.


The area of circle of the radius R  is  S = pi%2AR%5E2.


The ratio you are asking for is  S%2FS%5Bn%5D = pi%2F%28n%2Atan%28pi%2Fn%29%29

See the lesson
    - Area of a regular n-sided polygon via the radius of the inscribed circle
in this site.


Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!

R= radius of the circle = height of each of the 6 triangles forming the hexagon.
pi%2AR%5E2= area of the circle
2sqrt%283%29R%5E2= area of the hexagon
is the ratio asked for.

You may have a formula to calculate the area of a regular hexagon
as a function of its apothem (which in this case is R),
but I calculated it by figuring the sides and area of the triangles.
The angles of those triangles measure 60%5Eo .
That makes sin%2860%5Eo%29=sqrt%283%29%2F2=R%2Fside for the triangles,
so side=2R%2Fsqrt%283%29=2sqrt%283%29R%2F3 .
Area for 1 triangle would be
%281%2F2%29%282sqrt%283%29R%2F3%29R=sqrt%283%29R%5E2%2F3 ,
and the area of the hexagon is 6 times that
6%28sqrt%283%29R%5E2%2F3%29=2sqrt%283%29R%5E2 .