Find the area of the region between a square and an inscribed circle, when the square has a side length of 6 in.

radius of circle = 3 inches

area of circle = 9π
area of square = 36

area "between" = 36-9π = ...... (you decide on the decimals)

To find the area of the region between the square and the inscribed circle, we first need to determine the areas of both the square and the inscribed circle separately.

1. Area of the square:
The area of a square is given by the formula: Area = side length squared.
Given that the side length of the square is 6 inches, we can calculate the area by squaring the side length: Area = 6^2 = 36 square inches.

2. Area of the inscribed circle:
The inscribed circle is a circle that is tangent to all four sides of the square. The circle is inscribed in the square, so it has a diameter equal to the side length of the square.
The area of a circle is given by the formula: Area = π * radius^2.
Since the diameter of the circle is equal to the side length of the square, the radius of the circle is half the side length: radius = 6 / 2 = 3 inches.
Now we can calculate the area of the inscribed circle using the radius: Area = π * 3^2 = 9π square inches.

3. Finding the area between the square and the circle:
To find the area between the square and the circle, we subtract the area of the inscribed circle from the area of the square.
Area between the square and the circle = Area of the square - Area of the inscribed circle
= 36 square inches - 9π square inches
≈ 36 square inches - 28.27 square inches (using an approximation of π as 3.14)
≈ 7.73 square inches.

Therefore, the area of the region between the given square and the inscribed circle is approximately 7.73 square inches.