Acircle is inscribed a in a square. the side length of the square is x in. if the area of the shaded region is 20 pie, what is the radius of the circle?

8.555 in.

Step-by-step explanation:

So the area of the square is x^2.

This makes that the area of the circle is x^2 - 20 pie.

The radius of the circle is half of the diameter, so it is 0,5x.

The formula for the circle area is:

Area = pie * r^2.

x^2 - 20 pie = pie * (0,5x)^2

x^2 - 20 pie = pie * 0,25x^2

x^2 - pie * 0,25x^2 = 20 pie

x^2 * (1 - 0,25 pie) = 20 pie

x^2 = 20pie / (1 - 0,25 pie)

x = square root (20pie / (1 - 0,25 pie)) = 17.11

So the radius is 0,5 * 17.11 = 8.555 in.

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).