EQ: How do you find an arc length and an area of a circular sector? Section 3.2

Warm-Up/Activator What is the radius of the unit circle? What is the circumference (perimeter) of the unit circle? What is formula for the area of a circle of any radius r?

Arc Length A circle of any radius can be considered an expansion (size change by a factor of r) of the unit circle. In the unit circle, the radian measure of the central angle is equal to the length of the arc it intercepts. Therefore, to find an arc length of a circle radius r, we simply multiply the radian measure of the central angle by r. If we are given the angle measure in degrees, we must first convert to radians and then multiply by r.

Recall: Converting Degrees  Radians Convert 60° to radians. Give answer in terms of π.

Example 1 & Your Turn 1 In a circle with radius 10 centimeters, an arc is intercepted by a central angle with measure   . Find the arc length.\ In a circle with radius 15 inches, an arc is intercepted by a central angle with measure  . Find the arc length.

Example 2 & Your Turn 2 In a circle with radius 7.5 centimeters, an arc is intercepted by a central angle with measure 76˚. Find the arc length. Approximate the arc length to the nearest centimeter. In a circle with radius 20 meters, an arc is intercepted by a central angle with measure 113˚. Find the arc length. Approximate the arc length to the nearest meter.

Example 3 The International Space Station (ISS) is in an approximate circular orbit 400 kilometers above the surface of the Earth. If the ground station tracks the space station when it is within a 45˚ central angle of this circular orbit about the center of the Earth above the tracking antenna, how many kilometers does the ISS cover while it is being tracked by the ground station? Assume that the radius of the Earth is 6400 kilometers. Round to the nearest kilometer.

Your Turn 3 If the ground station in Example 3 could track the ISS within a 60˚ central angle of its circular orbit about the center of the Earth, how far would the ISS travel during the ground station tracking?

Example 4 Gears are inside many devices like automobiles and power meters. When the smaller gear drives the larger gear, then typically the driving gear is rotated faster than a larger gear would be if it were the drive gear. In general, smaller ratios of radius of the driving gear to the driven gear are called for when machines are expected to yield more power. The smaller gear has a radius of 3 centimeters, and the larger gear has a radius of 6.4 centimeters. If the smaller gear rotates 170˚, how many degrees has the larger gear rotated? Round the answer to the nearest degree.

Area of a Circular Sector The area of a circle is The area of the sector compared to the area of the whole circle is proportional to the measure of the central angle compared to a full rotation. Therefore, Radians Degrees

Example 5 & Your Turn 5 Find the area of the sector associated with a single slice of pizza if the entire pizza has a 14-inch diameter and the pizza is cut into 6 equal pieces. Find the area of a slice of pizza (cut into 8 equal pieces) if the entire pizza has a 16-inch diameter.

Example 6 & Your Turn 6 Sprinkler heads come in all different sizes depending on the angle of rotation desired. If a sprinkler head rotates 90˚ and has enough pressure to keep a constant 25-foot spray, what is the area of the sector of the lawn that gets watered? Round to the nearest square foot. If a sprinkler head rotates 130˚ and has enough pressure to keep a constant 30-foot spray, what is the area of the sector of the lawn it can water? Round to the nearest square foot.