1. 11.2 Areas of Sectors and circles

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

3. 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

4. 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
𝐴=𝜋 6.25
𝐴≈19.63
B. diameter of a circle with an area of square centimeters
113.1=𝜋 𝑟 2
113.1 𝜋 = 𝜋 𝑟 2 𝜋
36= 𝑟 2
36 = 𝑟 2
6=𝑟
Diameter = 12

5. 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

6. 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 =
𝑠𝑒𝑐𝑡𝑜𝑟 64𝜋 =
360𝑠=
360𝑠 360 =
𝑠≈39.10
Blue sector:
𝑠 𝜋 =
𝑠 64𝜋 =
360𝑠=
360𝑠 360 =
𝑠≈161.97

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

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

9. 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 )
𝑤𝑎𝑙𝑙=936− 32𝜋+256
𝑤𝑎𝑙𝑙=936−356.53
𝑤𝑎𝑙𝑙=579.47
𝑤𝑎𝑙𝑙=579 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡