Suppose large circle has radius r = 2, and area = A=πr^2 = 4π. (π is the symbol for pi).
Each smaller circle has radius r = 1, and area = π.
The small circles intersect each other at 90*.
If a line is drawn through the red area in a circle, it will be a chord and will create two segments, one in each overlapping circle. The formula for the area of a segment where the central angle is 90* is
½ ((πx90*/180*) - sin90*)) = ½(π/2 – 1). Thus each red area has a total area of 2(½(π/2 – 1) = π/2 – 1.
There are two of these red areas in each small circle. Therefore, the white (non-red) area in each small circle has an area = π – 2(π/2 – 1) = 2.
There are 4 of these white areas, and 4 of the red areas, so the total area of all of the white plus red areas is 4(2 + (π/2 – 1)) = 4+2π.
The blue area is the difference between the area of the large circle minus the area of the white and red areas.
Therefore the blue area is 4π – (4 +2π) = 2π – 4.
Since each red area is = π/2 – 1, the total red area is 4(π/2 – 1) = 2π – 4.
Thus the blue and red areas are equal.