The diagram consists of a large circle and four small circles.
Each of the smaller circles has radius 1 and touches the large circle and the two small circles next on either side of it.
What is the area of the shaded region?
Let O be the center of the circle.
Let A be the center of the smaller left-hand circle.
Let B be the center of the smaller top circle.
Let C be the center of the smaller right-hand circle.
Let D be the center of the smaller bottom circle.
ABCD is a square with each side = 2.
Therefore, the area of this square is 4.
Each of the smaller circle has an area of pi·12 = pi
However, one-fourth of each circle is contained in the center square, so each circle contributes
another ¾·pi of area to the white portion.
Total white area = 4 + 4(¾·pi) = 4 + 3·pi
The distance from B to D is the diagonal of the central square. Since each side of the central square
is 2, the diagonal is 2·sqrt(2).
So, the diameter of the outside circle is 2 + 2·sqrt(2) and its radius is 1 + sqrt(2).
The area of the outside circle is: pi·( 1 + sqrt(2) )2, making the area of the shaded section:
pi·( 1 + sqrt(2) )2 - (4 + 3·pi)