Essential Objectives: At the end of this lesson, you should be able to…  Find the length of an arc, given the measure of the central angle.  Find the area of a sector ______5.2 Length of an Arc & Area of a Sector Name: ______Date: ______

Key Concepts:

LENGTH OF AN ARC:

Def. A central angle of a circle is ______

______.

* If two central angles in different circles are congruent, the ratio of the length of their

intercepted arcs is ______.

Ex. Given Circle O and Circle Q and m BOA  m  DQC D

20 B 12

A 5 3 C O Q

Def. The length of any circular arc, s, is equal to the product of the measure of the radius of the circle, r, and the radian measure of the central angle,  , that it subtends.

Round to the nearest hundredth for the following examples. Ex 1. Find the length of an arc that subtends a central angle of 42 o in a circle of radius 8 cm.

Ex 2. Find the length of an arc that subtends a central angle of 38 o in a circle of radius 5 cm.

Ex 3. An arc is 7.5 cm long and subtends a central angle of 50 . Find the radius of the circle.

Ex 4. An arc is 12.3 cm long in a circle with a radius 7 cm. Find the degree measure of the central angle.

Ex 5. Brad and Bill are at point A on a circular track that has a radius of 150 feet, as show in the accompanying diagram. They run counterclockwise along the track from A to S, a distance of 247 feet. Find, to the nearest degree, the measure of the minor arc AS.

Ex 6. Find the distance along the road from point A to F.

AREA OF A SECTOR:

A sector of a circle is ______

______R T

θ S The ratio of the area of a sector to the area O r of a circle is equal to the ratio of its ______to the circumference.

Let A represent the area of the sector,

** If θ is the measure of the central angle expressed in radians and r is the measure of the radius of the circle, then the area of the sector, A, is as follows…

Example 1: A sector has an arc length of 16 cm and a central angle measuring 0.95 radians. Find the radius of the circle and the area of the sector.

Example 2: A sector has arc length 12 cm and a central angle measuring 1.25 radians. Find the radius of the circle and the area of the sector.

47 Example 3: A circle of radius 9 inches has a central angle of  . Determine the area of the sector 6 of the reference angle to the nearest hundredth of an inch.