題：以正方形的四個頂點為圓心,正方形的邊長為半徑劃四個圓.問四圓重疊區域的面積與正方形面積的比例是多少.

根據上面圖示，得

S = a^2

A = a^2 - 4D + 4B

B = a^2 - 2C + E

C = a^2 x pi / 4

D = a^2 - a^2 x pi / 4

E = 2F - G

F = a^2 x pi / 6

G = a/2 x sqrt(a^2 - (a/2)^2)
  = a/2 x sqrt(3 x a^2 / 4)
  = a^2 / 4 x sqrt(3)

E = a^2 x pi / 3 - a^2 / 4 x sqrt(3)

B = a^2 - a^2 x pi / 2 + a^2 x pi / 3 - a^2 / 4 x sqrt(3)

A = a^2 - 4(a^2 - a^2 x pi / 4) + 4(a^2 - a^2 x pi / 2 + a^2 x pi / 3 - a^2 / 4 x sqrt(3))

A = a^2 - 4 x a^2 + a^2 x pi + 4 x a^2 - 2 x a^2 x pi + 4 x a^2 x pi / 3 - a^2 x sqrt(3)
  = a^2 x (1 - 4 + pi + 4 - 2 x pi + 4 x pi / 3 - sqrt(3))
  = a^2 x (1 + pi/3 - sqrt(3))

r = A / S = 1 + pi/3 - sqrt(3)

為了清楚起見，步數稍為煩瑣了一點。