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Evaluate mentally.
A sector of a circle is the region enclosed by two radii.
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For each circle, find the area of the shaded sector and the length of the arc that outlines the sector. All units are centimeters. Give your answers in terms of .
Mai says, “I know how to find the area of a sector or the length of an arc for central angles like 180 degrees or 90 degrees. But I don’t know how to do it for central angles that make up more complicated fractions of the circle.”
A sector of a circle is the region enclosed by two radii. To find the area of a sector, start by calculating the area of the whole circle. Divide the measure of the central angle of the sector by 360 to find the fraction of the circle represented by the sector. Then multiply this fraction by the circle’s total area. We can use a similar process to find the length of the arc lying on the boundary of the sector.
The circle in the image has a total area of square centimeters, and its circumference is centimeters. To find the area of the sector with a 225-degree central angle, divide 225 by 360 to get , or 0.625. Multiply this by to find that the area of the sector is square centimeters. The length of the arc defined by the sector is because .
A sector is the region inside a circle between two radii.
In this diagram, the shaded region shows one sector of the circle.