Area of a Sector•Section 4.3954.3 Area of a SectorrθFigure 4.3.1In geometry you learned that the area of a circle of radiusrisπr2. Wewill now learn how to find the area of asectorof a circle.A sector isthe region bounded by a central angle and its intercepted arc, such as theshaded region in Figure 4.3.1.Letθbe a central angle in a circle of radiusrand letAbe the area of itssector. Similar to arc length, the ratio ofAto the area of the entire circleis the same as the ratio ofθto one revolution. In other words, again usingradian measure,area of sectorarea of entire circle=sector angleone revolution⇒Aπr2=θ2π.Solving forAin the above equation, we get the following formula:In a circle of radiusr, the areaAof the sector inside a central angleθisA=12r2θ,(4.5)whereθis measured in radians.Example4.8Find the area of a sector whose angle isπ5rad in a circle of radius 4 cm.Solution:Usingθ=π5andr=4 in formula (4.5), the areaAof the sector isA=12r2θ=12(4)
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