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?

Guest Jul 4, 2020

#1**0 **

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·1^{2} = 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)

pi

geno3141 Jul 5, 2020