Consider a square whose sides are of length x. Draw the largest circle that is completely contained within the square. Find the area of the region that is inside the square and but not inside the circle. Express your answer in terms of x and any other mathematical constants that may be necessary.

A₁ = x²* ( 1 -  π /2)  square units

Step-by-step explanation:

Let´s call the area of the region that is inside the square but not inside the circle A₁

And the area of the square A(s)

And the area of the circle A(c)

Then  the area we are looking for is:

A₁  =  A(s) - A(c)

If the circle has a radius   r then

A(c) = π*r²

A(s) = x²

In the square diagonal of the square is d = √ x² + x²  ;  d = √2 * x

Half of this distance is  d/2 = (√2 /2)*x

There is a right triangle with points:

o The center of the square (which is the center of the circle)

a  Any of the corners of the square

b  Middle point of a side of the square

According to that

r the radius of the circle is:

r² + (x/2)²  = (d/2)²

r² + (x/2)²  = [ (√2 /2)*x]²

r²   =  (2/4)*x² - x²/4

r²   = (x²/2) - x² / 4

r²   = (1/2)*x²

r  =  x/√2

Then the area of the circle is

A(c) = π * r²

A(c) =  π * x²/2

And  A₁  = x² -  π * x²/2

A₁ = x²* ( 1 -  π /2)  square units  or   A₁ =[ ( 2 -  π)/2 ]*x²

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