Question

A circle with radius 9 has a sector with a central angle of 120°.

Answer 1

120º is 1/3 of a complete revolution of 360º. So the area of this sector should be 1/3 the area of the complete circle.

A circle with radius 9 has area 9^2 π = 81π.

So the sector has area 81π/3.

Put another way: The area A of a circular sector and its central angle θ (in degrees) occur in the same ratio as the area of the entire circle with radius r according to

A / θ º = (π r ^2) / 360º

==>  A = π/360 θ r ^2

In this case, r = 9 and θ = 120º, so

A = π/360 * 120 * 81 = 81π/3