Four Circle Problem

[bob012345]

TL;DR

Determine the fraction of the darker area to the total area of the figure.

This is a little exercise for fun. Determine the area of the darker petals to the total area. The four circles are identical. Have fun!

 

[.Scott]

Each half a petal is bounded by a chord and a 90-degree slice of the circle. So the area would (pi (r^2)/4) - (r^2)/2 = (r^2)(pi/4 - 1/2). Since there are 8 of those, the total petal area is 8(r^2)(pi/4 - 1/2)=(2pi-4)(r^2).
The total area is 4 circles minus the overlap: 4pi(r^2)-(2pi-4)(r^2) = (2pi+4)(r^2).
Dark petals to total area is: (2pi-4)(r^2)/(2pi+4)(r^2) = (2pi-4)/(2pi+4) = (pi-2)/(pi+2)

 

[bob012345]

Spoiler

I got $(\pi-2)/(\pi+2)$