Section 7-2 Sectors of Circles

Sector A sector of a circle, shaded in red below, is the region bounded by a central angle and the intercepted arc. Your geometric intuition should tell you that the length s of the intercepted arc is some fraction of the circumference of the circle and that the area of the sector is the same fraction of the area of the circle.

Sample For example, suppose the central angle of the sector is 60˚ and the radius is 12. Then the arc length of the sector is of the whole circumference

Sample Similarly, the area of the sector is of the area of the whole circle, or

Formulas In general, we have the following formulas for the arc length s and area K of a sector with central angle Θ. If Θ is in degrees, then: (1) (2)

Formulas If Θ is in radians, then: (1a) s = r Θ (2a) k =

Formulas Notice that formulas (1a) and (2a) are more straightforward than formulas (1) and (2). One reason for using radian measure is that many formulas in calculus are expressed more simply in radians rather than degrees. By combining formulas (1a) and (2a), we can obtain a third area formula: So, (2b)

Formulas By combining formulas (1a) and (2a), we can obtain a third area formula: So, (2b)

Apparent Size When there is nothing in our field of vision against which to judge the size of an object, we perceive the object to be smaller when it is farther away.

Apparent Size So, how big an object looks depends not only on the object’s size, but also on the angle that it subtends at our eyes. The measure of this angle is called the object’s apparent size.

Apparent Size Use this diagram when you solve apparent size problems. Θ s

Apparent Size Θ is the angle measured in degrees or radians R is the distance between you and the object or the two objects S is the diameter of the round object or the height of a non-round object

Example A sector of a circle has radius 4 cm and central angle 45°. Find its arc length and area.

Example A sector of a circle has radius 6 cm and central angle . Find its arc length and area.

Additional Example A sector has perimeter 16 cm and area 15 square centimeters. Find its radius r and arc length s.

Additional Example: A phonograph record with diameter 12 in. turns at rpm. Find the distance that a point on the rim travels in one minute