11.2 Areas of Sectors and circles

What we will learn Use area of a circle Population density Area of sectors Use area of sectors

Needed vocab Population density: measure of how many people live within a given area. Sector of a circle: region bounded by two radii of the circle and their intercepted arc

Ex. 1 Using area formula of a circle Area of a circle: 𝐴=𝜋 𝑟 2 Find each measure: A. area of a circle with a radius 2.5 centimeters 𝐴=𝜋 2.5 2 𝐴=𝜋 6.25 𝐴≈19.63 B. diameter of a circle with an area of 113.1 square centimeters 113.1=𝜋 𝑟 2 113.1 𝜋 = 𝜋 𝑟 2 𝜋 36= 𝑟 2 36 = 𝑟 2 6=𝑟 Diameter = 12

Ex. 2 Using population density 𝑑𝑒𝑛𝑠𝑖𝑡𝑦= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑙𝑎𝑛𝑑 A. About 430,000 people live in a 5 mile radius of a city’s town hall. Find the population density. 𝑑= 430,000 𝜋 5 2 𝑑= 430,000 25𝜋 𝑑≈5475 people per square mile B. A region with a 3 mile radius has a population density of 6195 people per square mile. Find the number of people in the area. 6195= 𝑝 𝜋 3 2 6195= 𝑝 9𝜋 6195 9𝜋 =𝑝 175,159≈𝑝

Ex. 3 Area of sectors Area of a sector: 𝑠𝑒𝑐𝑡𝑜𝑟 𝜋 𝑟 2 = 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 360 Find area of red and blue sectors Red sector: 𝑠𝑒𝑐𝑡𝑜𝑟 𝜋 8 2 = 70 360 𝑠𝑒𝑐𝑡𝑜𝑟 64𝜋 = 70 360 360𝑠=14074.335 360𝑠 360 = 14074.335 360 𝑠≈39.10 Blue sector: 𝑠 𝜋 8 2 = 290 360 𝑠 64𝜋 = 290 360 360𝑠=58307.96 360𝑠 360 = 58307.96 360 𝑠≈161.97

Your Practice Find area of each sector: Blue: Red: 𝑠 196𝜋 = 240 360 𝑠 196𝜋 = 120 360 360𝑠=73890.26 360𝑠 360 = 73890.26 360 𝑠=205.25 Blue: 𝑠 196𝜋 = 240 360 360𝑠=147,780.52 360𝑠 360 = 147,780.52 360 𝑠≈410.50

Ex. 4 using the area of a sector Find the area of the circle Find 𝑟 2 first 35 𝜋 𝑟 2 = 40 360 40𝜋 𝑟 2 =12600 40𝜋 𝑟 2 40𝜋 = 12600 40𝜋 𝑟 2 =100.27 Plug into area or circle equation 𝐴=𝜋 100.27 𝐴=315

Ex. 5 Finding area of a region You want to paint the wall. To the nearest square foot, what is the area of the region you need to paint? Area of wall = rectangle – (half circle + square) 𝑤𝑎𝑙𝑙=36 26 −( 𝜋 8 2 2 +16 16 ) 𝑤𝑎𝑙𝑙=936− 32𝜋+256 𝑤𝑎𝑙𝑙=936−356.53 𝑤𝑎𝑙𝑙=579.47 𝑤𝑎𝑙𝑙=579 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡