Answer:
[tex]31.73\ in^2[/tex]
Step-by-step explanation:
1. Approach
The easiest way to solve this problem is to find the area of the triangle and the area of the inscribed circle. Then subtract the area of the inscribed circle from the area of the triangle.
2. Find the area of the triangle,
The area of the triangle can be found using the following formula,
[tex]A_t=\frac{b*h}{2}[/tex]
Where (b) represents the base of the triangle, and (h) represents the height.
Substitute in the given values and solve,
[tex]A_t=\frac{b*h}{2}\\\\A_t=\frac{12*10}{2}\\\\A_t=\frac{120}{2}\\\\A_t=60[/tex]
3. Find the area of the inscribed circle,
The formula to find the area of a circle is the following,
[tex]A_c=(pi)r^2[/tex]
Where (pi) represents the numerical value (3.1415...) and (r) represents the radius of the circle. By its definition, the radius of a circle is always half of the diameter, thus the radius of the given circle is (3).
Substitute in the given values and solve,
[tex]A_c=(pi)r^2\\\\A_c=(pi)(3)^2\\\\A_c=(pi)9\\\\A_c=28.26[/tex]
4. Find the area of the shaded region,
To accomplish this task, subtract the area of the circle from the area of the triangle,
[tex]A_t-A_c=A_s[/tex]
[tex]60-28.26=A_s[/tex]
[tex]31.73=A_s[/tex]
