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. .