Definition: A sector is the region of a circle bounded by two radii and the arc between them; its area is the fraction (angle / 360) times pi r^2.
What is a sector?
Picture a slice of pizza: two straight edges that are radii, and a curved edge that is an arc. A semicircle is a sector with a 180 degree angle.
How do you find its area?
Area = (θ / 360) · πr2. In radians, A = (1/2) r2θ. The idea is the same as for arc length: take a fraction of the whole.
Example. A sprinkler waters a sector of radius 8 m with a central angle of 110 degrees. The area is (110/360)(π)(64) = (11/36)(64π), which is about 61.4 square meters. The perimeter of the sector is the arc plus two radii: (110/360)(2π)(8) + 16, about 15.36 + 16 = 31.4 m.
What mistakes are frequent?
Using the circumference formula for area, or forgetting to square the radius. Another is leaving out the two radii when asked for the perimeter. Area uses square units, and arc length uses plain units, so the units can help you check.