We can construct a rifght triangle with legs of (5 - r) and a hyotenuse of (5 + r) where r is the radius of the smaller circle
So by the Pythagorean Theorem we have
2 ( 5 - r)^2 = (5 + r)^2 simplify
2 ( r^2 - 10r + 25) = r^2 + 10r + 25
2r^2 - 20r + 50 =r^2 + 10r + 25
r^2 - 30r + 25 = 0 complete the square on r
r^2 - 30r + 225 = -25 + 225
(r - 15)^2 = 200 take the smaller root
r - 15 = -sqrt (200)
r = 15 - 10sqrt (2)
So the blue area = (area of quarter square -area of quarter circle -area of small circle ) =
10^2/4 - (25/4)pi - (15 - 10sqrt (2))^2*pi ≈ 3.05 units^2