11.1.3 Sector Area and Arc Length (continued) Example 3 Finding the Area of a Segment
A segment is the area bounded by an arc and its chord. The area of a segment is equal to the area of a sector minus the area of the triangle formed by the chord and the radii that define the sector. The area of a segment is determined in this example. The diameter and the measure of the central angle are given. To determine the area of the segment, begin by finding the area of the sector defined by the central angle. Use the formula for sector area. Substitute the values for radius and the measure of the central angle into the formula. Solving for area yields A = 54 π in 2 . Find the area of the triangle by using the formula for the area of a triangle. The central angle of the triangle is 60 ° and the height of the triangle bisects the triangle into two 30 ° -60 ° -90 ° right triangles whose shorter legs are 9 inches long, (1/2)18 in., and whose longer leg, the height, is 9 3. Substituting into the formula for the area of a triangle gives 81 3 in . 2 The area of the segment is the difference between the two areas, which is approximately 29.35 in 2 . The length of an arc is determined in this example. The radius and the measure of the central angle are given. To determine the length of the arc, use the formula for the length of an arc. Substitute the values for radius and the measure of the central angle into the formula. Solving for length yields L = (4/3) π in.
Example 4 Finding Arc Lengths
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