Final answer:
To find the shaded area, first find the radius of the circle using the given arc length and central angle. Then, calculate the area of the circle and the area of the triangle formed by the radii. Finally, subtract the area of the triangle from the area of the circle to find the shaded area, which is expressed as a fraction times pi.
Step-by-step explanation:
To find the shaded area, we need to first find the radius of the circle. We are given that the length of arc FG is πFG = ⅟6π and the length of EF is 3. Since arc length is given by the formula arc length = radius * central angle, we can set up a proportion to find the radius of the circle:
πFG = ⅟6π = (radius) * (central angle) = r * 3
Simplifying the equation, we have r = ⅟6. Now, we can find the shaded area:
The shaded area is the difference between the area of the circle and the area of the triangle formed by the radii EF and FG:
Area of circle = πr^2 = π(⅟6)^2 = ⅟36π
Area of triangle = (1/2) * EF * FG = (1/2) * 3 * ⅟6 = ⅟9
Therefore, the shaded area is ⅟36π - ⅟9 = ⅕36π = ⅕36π