Find the area of the blue shaded region...

Question

A circle is inscribed in a square as shown in figure. A smaller circle is drawn tangent to two sides of the square and externally tangent to the inscribed circle. Find the area of the blue shaded region...

Answer 1

Answer 2

The small circle's radius is

2(5-r)²= (5+r)²
√2 (5-r) = 5+r
5√2 - √2.r = 5+r
(1+√2)r =5√2 - 5
R = (5√2 - 5)/(1+√2)
The area of the shaded space is
5²-(25pi/4)-((5√2-5/1+√2)²pi)

Answer 3

The problem lies in finding the small circle area; since its radius doesn't start from the square corner, we cannot just subtract the Bigger circle radius from the square diameter and divide by 2.

The solution i came with is finding the ratio between (the the bigger circle radius) and (the part of the diameter that is crossing over the blue shaded area + circle radius)...
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This will let us know how to calculate the small circle radius by knowing only the length of the line that is crossing over the blue shaded area.

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So the shaded section area (inlc small circle) :
(10*10-pi*5^2)/4)

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small circle area:
pi*(((5sqrt(2)-5)-(5sqrt(2)-5)*(5sqrt(2)-5)/(10+(5sqrt(2)-5)))/2)^2

=
3.053
.

This is the calculation I came up with:
http://bit.ly/2mbAFqt