Let's solve the problem step by step:
1. Calculate the Diagonal of the Square:
- Given the side length of the square is [tex]\(6\)[/tex] inches.
- In a square, the diagonal can be found using the ratio [tex]\(1:1:\sqrt{2}\)[/tex] because each half-angle of the square forms a right triangle.
- Utilizing this ratio, the diagonal [tex]\(d\)[/tex] of the square is:
[tex]\[
d = 6 \times \sqrt{2}
\][/tex]
2. Calculate the Radius of the Circle:
- The diagonal of the square equals the diameter of the circle.
- Hence, the diameter [tex]\(D\)[/tex] of the circle is:
[tex]\[
D = 6\sqrt{2}
\][/tex]
- The radius [tex]\(r\)[/tex] of the circle is half of the diameter:
[tex]\[
r = \frac{6\sqrt{2}}{2} = 3\sqrt{2}
\][/tex]
3. Calculate the Area of the Circle:
- The formula for the area of a circle is [tex]\(A = \pi r^2\)[/tex].
- With the radius [tex]\(r = 3\sqrt{2}\)[/tex], the area of the circle [tex]\(A_{\text{circle}}\)[/tex] is:
[tex]\[
A_{\text{circle}} = \pi (3\sqrt{2})^2 = \pi \times 18 = 18\pi \text{ square inches}
\][/tex]
4. Calculate the Area of the Square:
- The area of a square is given by side length squared:
[tex]\[
A_{\text{square}} = 6^2 = 36 \text{ square inches}
\][/tex]
5. Calculate the Area of One Segment:
- The total area outside the square but inside the circle is:
[tex]\[
A_{\text{circle}} - A_{\text{square}} = 18\pi - 36
\][/tex]
- There are four segments formed by the square inside the circle. Therefore, the area of one segment is:
[tex]\[
\text{Area of one segment} = \frac{18\pi - 36}{4} = \frac{18\pi}{4} - \frac{36}{4} = \frac{9\pi}{2} - 9
\][/tex]
Thus, the area of one segment formed by the square inscribed in the circle is:
[tex]\[
A = \frac{9\pi}{2} - 9 \text{ square inches}
\][/tex]
