Pentastar!
Consider a regular pentagon, whose vertices are connected to the center. In each isosceles triangle, we draw the incircle. These incircles are tangential to the same point, and this forms a curved "pentastar" in the center.
How does the area of the central "pentastar" compare to the area of one of these circles?
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2 solutions
Without the loss of generality, let the radius of the circle be 1. The two tangents on a circle are at an angle of 360/5=72. This is cyclic a kite, with Center of the circle, two points of tangency, and the meeting point of the two tangents meet., In this kite, the radii are at 180-72 =108. So the length of the tangent is Tan54. Resulting area of the kite is 2 * 0.5 * Tan54 * 1.
But this includes the segment of the circle that makes an angle of 108 at the circle's center. So the net area of the pentastar =
5
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.
Moderator note:
I do not understand what you're trying to express here. You seem to be defining the options, as opposed to presenting a solution to the problem.