Finding Arc Lengths and Areas of Sectors

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Presentation transcript:

Finding Arc Lengths and Areas of Sectors Lesson 35 Finding Arc Lengths and Areas of Sectors

Arc Length - To find the length of an arc, use this formula, where m is the degree measure of the arc. 𝐿=2𝜋𝑟 𝑚° 360°

Example 1 Finding Arc Length Find each arc length. Give your answer in terms of π. a. Find the length of 𝑋𝑌 . SOLUTION 𝐿=2𝜋𝑟 𝑚° 360° 𝐿=2𝜋(15) 60° 360° L = 5π cm

Example 1 Finding Arc Length Find each arc length. Give your answer in terms of π. b. Find the length of an arc with a measure of 75° in a circle with a radius of 4 feet. SOLUTION 𝐿=2𝜋𝑟 𝑚° 360° 𝐿=2𝜋4 75° 360° 𝐿= 5 3 𝜋𝑓𝑡

Sector of a Circle - The region inside a circle bounded by two radii of the circle and their intercepted arc. Area of a Sector - To find the area of a sector (A), use the following formula, where r is the circle’s radius and m is the central angle measure: 𝐴=𝜋 𝑟 2 𝑚° 360°

Example 2 Finding the Area of a Sector Find the area of each sector. Give your answer in terms of π. a. Find the area of sector XOY. SOLUTION 𝐴=𝜋 𝑟 2 𝑚° 360° 𝐴=𝜋 12 2 95° 360° 𝐴=38𝜋 𝑖𝑛 2

Example 2 Finding the Area of a Sector Find the area of each sector. Give your answer in terms of π. b. Find the area of a sector with an arc that measures 174° in a circle with a radius of 13 meters. 𝐴=𝜋 𝑟 2 𝑚° 360° 𝐴=𝜋 (13) 2 174° 360° 𝐴= 4901 60 𝜋 𝑚 2

Example 3 Solving for Unknown Radius Find the radius of the circle to the nearest hundredth of a meter. SOLUTION Substitute the known measures into the formula for the area of a sector, then solve for r. 𝐴=𝜋 𝑟 2 𝑚° 360° 100=𝜋 𝑟 2 246° 360° 110∙360 246𝜋 = 𝑟 2 r ≈ 6.83 m

Example 4 Solving for Unknown Central Angle Find the central angle measure of 𝑅𝑆 to the nearest hundredth of a degree, if the length of the arc is 12 centimeters. SOLUTION 𝐿=2𝜋𝑟 𝑚° 360° 12=2𝜋(14) 𝑚° 360° 12∙360 2∙14∙𝜋 =𝑚° m° ≈ 49.11°

Example 5 Application: Farming\ A spray irrigation system has a radius of 150 feet. If it rotates through a 175° central angle, what is the area that the system covers? Round your answer to the nearest square foot. SOLUTION 𝐴=𝜋 𝑟 2 𝑚° 360° 𝐴=𝜋 150 2 175° 360° 𝐴≈34361 𝑓𝑡 2

You Try!!! a.Find the length of an arc with a measure of 125° in a circle and 12-mile radius. Round to the nearest hundredth of a mile. c.Find the radius to the nearest hundredth of a centimeter.

Assignment Page 226 Lesson Practice (Ask Mr. Heintz) Page 227 Practice 1-30 (Do the starred ones first)