600 CHAPTER 6 The Circular Functions and Their Graphs Area of a Sector The area of a sector of a circle of radius r and central angle u is given by the following formula. = 1 2 r2 U, where U is in radians CAUTION As in the formula for arc length, the value of U must be in radians when this formula is used to find the area of a sector. Center-pivot irrigation system EXAMPLE 7 Finding the Area of a Sector-Shaped Field A center-pivot irrigation system provides water to a sector-shaped field with the measures shown in Figure 11. Find the area of the field. SOLUTION First, convert 15° to radians. 15° = 15 a p 180b = p 12 radian Convert to radians. Now find the area of a sector of a circle. = 1 2 r2 u Formula for area of a sector = 1 2 132122 a p 12b Let r = 321 and u = p 12 . ≈13,500 m2 Multiply. 158 321 m Figure 11 S Now Try Exercise 111. 6.1 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the of the circle has measure 1 radian. 2. 360° = radians, and 180° = radians. 3. To convert to radians, multiply a degree measure by radian and simplify. 4. To convert to degrees, multiply a radian measure by and simplify. CONCEPT PREVIEW Work each problem. 5. Find the exact length of the arc intercepted by the given central angle. 4 P 2 6. Find the radius of the circle. 6P 3P 4
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