find the area of one segment formed by a square with sides of 6" inscribed in a circle.

To find the area of one segment formed by a square inscribed in a circle, we need to use a formula that involves both the side length of the square and the radius of the circle.

Here's how you can calculate the area of the segment:

1. Start by finding the radius of the circle. Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.
- The diagonal of a square can be calculated using the formula: diagonal = side length * √2.
- For our square with a side length of 6 inches, the diagonal is 6 * √2 inches.
- Since the diameter of a circle is twice the radius, the radius of the circle is ((6 * √2) / 2) inches, which simplifies to 3√2 inches.

2. Next, calculate the area of the sector formed by the segment. A sector is a fraction of the whole circle, and its area can be calculated using the formula: area of sector = (θ/360) * π * r^2.
- The angle θ of the sector can be calculated by finding the angle formed by one of the corners of the square and the center of the circle. This angle can be found using trigonometry. In this case, the angle θ is equal to 2 * arctan(side length / (2 * radius)).
- Plugging in the values, we get θ = 2 * arctan(6 / (2 * 3√2)).
- Using a calculator, the value of θ is approximately 74.83 degrees.
- Now, we can calculate the area of the sector using (74.83/360) * π * (3√2)^2.

3. Finally, calculate the area of the triangle formed by the segment. Since the square divides the segment into two equal triangles, we can find the area of one triangle and then multiply it by 2.
- The area of a triangle can be calculated using the formula: area of triangle = (base * height) / 2.
- In this case, the base of the triangle is the side length of the square (6 inches), and the height can be found using Pythagorean theorem as the square root of (radius^2 - (side length/2)^2).
- Plugging in the values, the height of the triangle is √((3√2)^2 - (6/2)^2).

4. Add the areas of the sector and the triangle to get the total area of one segment.

Let me calculate the values for you.

Area of square = 6² in² = 36 in²

radius of circle
r = (1/2)square diagonal
= (1/2)√(6²+6²)
= (1/2)√(72)
= √(18)

Area of circle = πr²
= 18π

Area of one of the 4 sectments
= (Area of circle - area of square)/4