11.2 Area of Sector & Population Density

How many squares are in the circle? AREA The area of a circle is the number of unit squares that can fit into the circle. How many squares are in the circle?

Example: Find the Area of the Circle 2) Given the area of the circle, find the radius. A = 380 in2 r 380 = 𝜋𝑟2 380 𝜋 = 𝑟2 cm2 11.0 in ≈𝑟 cm2

SECTOR sector of a circle: part of the interior of a circle bounded by two radii and an arc. The size of a slice of pizza.

AREA OF A SECTOR = (fraction of the circle) X (area of circle) The fraction of the circle multiplied by the area of the circle. Area of a Sector = (fraction of the circle) X (area of circle) = x C = x 𝜋r2

Examples: 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟= 𝑎 360 ∙𝐴 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟= 𝑎 360 ∙𝐴 1) 2) 72° 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟= 𝑎 360 ∙𝐴 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟= 𝑎 360 ∙𝐴 = 72 360 ∙𝜋82 𝐴𝑟𝑒𝑎 𝑟𝑒𝑑 = 120 360 ∙𝜋142 = 64 5 𝜋 𝑖𝑛2 = 196 3 ∙𝜋 𝑓𝑡2 ≈40.2 𝑖𝑛2 ≈205.3 𝑓𝑡2

Example 3: The wall shown has a doorway cut out of it. Jamie needs to paint the wall. Find, to the nearest tenth of a foot, the area to be painted. The area to be painted is a rectangle – the entrance area. The entrance can be divided into a semi-circle and a square. Area(paint) = A(rectangle) – [A(square + A(semi-circle)] = 36∙26 – (16∙16+ 180 360 𝜋∙82) = 936 –(256+32𝜋) ≈579.5 𝑓𝑡 2

Population Density The population density of a city, parish or state is a measure of how many people live within a given area. Population density = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑙𝑎𝑛𝑑

Examples: 4) About 58,000 people live in a region with a 2 mile radius. Find the population density. 5) A region with a 3 mile radius has a population density of 1000 people per square mile. Find the number of people who live in the region. Pop. density = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑙𝑎𝑛𝑑 ≈ 58,000 4𝜋 about 4615 people per mi2 Pop. density = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑙𝑎𝑛𝑑 1000 = # 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 9𝜋 9000 𝜋 = # of people about 28,274 people