Four quarter circles are drawn inside a unit square, as shown below. Find the area of the shaded region.
+128475 See the following :
Using the Law of Cosines we can find EF = one side of a square which comprises part of the shaded region
EF = sqrt [ BE^2 + BF^2 - 2 ( BE * BF)cos (30°) ]
EF = sqrt [ 1^2 + 1^2 - 2 (1 * 1)sqrt (3/2) ]
EF = sqrt ( 2 - sqrt (3) )
EF^2 = area of the square = (2 -sqrt(3) )
And the area between the circle and the side of the square =
Area of 30° sector of the circle = (1/12) pi 1^2 = pi/12 less
Area of triangle BEF =(1/2) (1)^2 sin (30°) = (1/2) * (1/2) = 1/4
So this area = pi/12 - 1/4 = [ pi - 3] / 12
And we have 4 of these regions = [pi -3 ] / 3 = pi/3 -1
So....the total shaded area = (2 -sqrt (3)) + (pi/3 -1) =
[1 + pi/3 - sqrt (3)] units^2 ≈ .315 units^2
