Geometry Question

Question

A regular pentagon ABCDE, each of length 4 is drawn, and point F is the center of the pentagon. From point F, 5 circles are drawn such that the radius of each is the distance between point F and each vertex of the pentagon. Let P be the set of all areas that are bounded by any three adjacent circles. What is the area of P ?

Answer 1

Since no one else has answered I will describe the solution.

Call the 3 circles A, Band C.

The circles overlap.

The area required is area of A + area of B + area of C - (area where A and B overlap + area where B and C overlap) + area where all 3 circles overlap.

Of course, the overlap areas are all lenticular and can be calculated...but the calculations are messy at best.

Answer 2

I didn't finish calculating the area, but I created this cool diagram from the description using Geogebra so that you could visualize it better.

https://www.geogebra.org/geometry/kfxg8dtp

Now its just a matter of calculating angles and lengths, so that you can find area of sectors.