SOLUTION: In a circle with a diameter of 12 inches, a regular five-pointed star is inscribed. What is the area of that part not covered by the star? Please explain.

Question 1113151: In a circle with a diameter of 12 inches, a regular five-pointed star is inscribed. What is the area of that part not covered by the star? Please explain.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The regular five-pointed star,
and the circle with radius where it is inscribed,
would look like this:

We see that the star is a pentagon plus 5 isosceles triangles.
If you were given a formula to calculate areas of regular n-pointed stars inscribed in circles,
by all means use those formulas.
I think it is better not to use a formula if you do not have the formulas,
and are not able to deduce one quickly on the spot.
I can deduce one, but I prefer not to let formulas interfere with visualizing and understanding, so I will show a different approach.

We can figure out some angles:

The angle between the top point of the star and point (the rightmost point of the 5-pointed star)
measures ,
and the complementary angle,
the one between the positive x-axis and the red ray going through ,
measures .
With that we can calculate as approximately .
That is (in inches) the apothem of the pentagon, while the height (in inches) of the isosceles triangles is about
.
Now we only need the of the pentagon,
and then we would be able to calculate the area of the pentagon as
,
and the area of the 5 triangles that make the points of the star as
.
There must be other formulas that we could look up,
but knowing that the angle marked with a green arc
(half of the central angles of the pentagon),
measures
we can calculate the length (in inches) of half a side of the pentagon as

That makes the perimeter of the pentagon approximately .
The area of the pentagon (in square inches) is
.
The area (in square inches) of the five triangles forming the tips of the star is
.
The area (in square inches) of 5-pointed star is
.

The area (in square inches) of the circle is ,
so the area of that part not covered by the star is

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