5) A square is inscribed in a circle. If the area of the circle is 36 , what is the circumference of a circle? 6a) Find the perimeter of the square. b) Find the ratio of the shaded region to the area of the square.

To find the circumference of the circle, we first note that the area of the circle is given by the formula \( A = \pi r^2 \). Given that the area is 36, we can set up the equation \( \pi r^2 = 36 \). Solving for \( r \) gives us \( r = \sqrt{\frac{36}{\pi}} \). The circumference \( C \) can then be calculated using the formula \( C = 2 \pi r \), which leads to \( C = 2\sqrt{36\pi} = 12\sqrt{\pi} \). Next, for the perimeter of the square, we find the relationship between the square and the circle. The diameter of the circle is the diagonal of the square, which is \( d = 2r \). The side length \( s \) of the square can be derived from the equation \( s\sqrt{2} = d \). Substituting \( r \) gives \( s = \sqrt{2} \cdot \sqrt{\frac{36}{\pi}} = 6\sqrt{\frac{2}{\pi}} \). Thus, the perimeter \( P \) of the square is \( P = 4s = 24\sqrt{\frac{2}{\pi}} \). For the ratio of the shaded region (the area of the circle minus the area of the square) to the area of the square, we first find the area of the square \( A_s = s^2 = \left(6\sqrt{\frac{2}{\pi}}\right)^2 = 72\cdot\frac{2}{\pi} = \frac{144}{\pi} \). The area of the circle is 36, so the shaded area is \( A_c - A_s = 36 - \frac{144}{\pi} \). The ratio is then \( \frac{36 - \frac{144}{\pi}}{\frac{144}{\pi}} = \frac{36\pi - 144}{144} = \frac{36\pi - 144}{144} = \frac{\pi - 4}{4} \).