Final answer:
To find the area of the sector, divide the central angle by 360 to get the fraction of the circle it represents. Then, multiply that fraction by the total area of the circle. In this case, the area of the sector is 27π.
Step-by-step explanation:
To find the area of the sector, we need to find the fraction of the circle that is represented by the central angle. The central angle of 120 degrees is equal to 120/360 = 1/3 of the total angle of a circle. So, the area of the sector is equal to 1/3 of the total area of the circle.
To find the area of the circle, we can use the formula A = πr². We know that the area of the circle is 81π, so we can set up the equation 81π = πr². Simplifying this equation, we get r² = 81. Taking the square root of both sides, we get r = 9.
Now, we can find the area of the sector by multiplying the area of the circle by the fraction of the circle represented by the central angle. The area of the sector is (1/3) * 81π = 27π.