Question

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Answer 1

Whenever you see a problem asking you for the shaded area, usually it involves subtracting the area of the shape that's completely inside from the area of the larger shape.
In this case, you have a smaller square completely inside a larger circle. To solve this problem, find the area of that circle and subtract the area of the square from it.

1) Remember that the area of a circle is $A= \pi r^{2}$, where A = area and r = radius of the circle. You know that r = 6, so plug that in and solve for A:
$A= \pi 6^{2}$
$A= 36 \pi$

2) Finding the area of the circle is probably the hardest part of this problem. If you look at the picture I attached, notice how the diameter of the circle, d, is  
also the diagonal of the square and the hypotenuse of the triangle formed by the diagonal cutting through the square.

Remember the Pythagorean theorem is $a^{2}+ b^{2} = c^{2}$, where a and b are the legs of the triangle and c is the hypotenuse. 

You already know the value of c because the c = diameter = 2(radius), so d=2(6)=12. You also know that a and b must also be the same value since it's the sides of a square, and a square has four equal sides. So, go ahead and put in a variable, "s" for side, or both a and b. Plug these into the Pythagorean theorem and solve for s:
$s^{2}+ s^{2} = 12^{2} \\ 2 s^{2} = 144\\ s^{2} = 72 \\ s = \sqrt{72}$

Now you know the side of the square, s, is equal to √72. Remember that the equation for the area of a square is: $A = s^{2}$, where s=side of the square. Plug s=√72 into the equation:
$A = s^{2}\\ A = \sqrt{72} ^{2}\\ A=72$

3) Now subtract the area of the square from the area of the circle to find the area of your shaded region: 
$36 \pi -72$ ≈ 41.1. 

The answer is C) 41.1.

Answer 2

Area of a circle = pi r^2
Area of a square = s^2 where s has to be derived from r.
pi = 3.14
r = 6

The center of the square is also the center of the circle.
The side of the square (draw this) is the hypotenuse of a right angle triangle which has the radius of the circle as the two smaller sides of the right angle.
r^2 + r^2 = s^2
6^2 + 6^2 = s^2
s^2 = 36 + 36
s^2 = 72 
Area of the square = 72 square units. You don't need to find s.

The area of the circle is pi r^2
Area_circle = 3.14 * 6 * 6
Area_ circle = 3.14 * 36
Area_circle = 113.04

The area of the shaded area = the area of the circle - the area of the square.
Area of the shaded area = 113.04 - 72 = 41.04

C <<<< answer
The answers are so close and if you round in the wrong place, you could get any one of them.

Sneaky!!!!