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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?

 

 Jul 4, 2020
 #1
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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)

     pi

 Jul 5, 2020

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