7.4: Arc Length & Area of a Sector

7.4: Arc Length & Area of a Sector

Objective: Use proportional reasoning to find arc length and area of a sector

Arc Length

• Arc Length = measurement of a portion of the circumference

• To find arc length, we will need to multiply circumference by the fraction of the circle the arc subtends

• Arc Length =   $2\pi r\left(\frac{\theta}{360}\right)$  

Ex 1: Find the length of the minor arc

• Arc Length =  $2\pi r\left(\frac{central\ angle}{360}\right)$  

• r = 12; central angle = 45

• Arc Length =  $2\pi\left(12\right)\left(\frac{45}{360}\right)$  

• Arc Length =  $3\pi$  or  $9.42$  

Ex 2:

• A pizza has been cut into 10 equal pieces. The outer edge of the crust of one piece measures 7 units. Find the diameter of the pizza to the nearest whole number.

• Arc Length = 7 units; Central Angle = (360/10) = 36

•  $7=2\pi r\left(\frac{36}{360}\right)$  

• r = 11.140846

• The diameter of the pizza is 22 units

Multiple Choice

A central angle measures $230\degree$ and its radius is 6 cm. What is the length of the arc intercepted by the angle? 

• 12.o4 cm

• 144.51 cm

• 24.08 cm

• 72.26 cm

Multiple Choice

What is the radius of a circle that has a central angle of 230 degrees and an arc length of 60 miles.

• 6.10 miles

• 74.95 miles

• 29.89 miles

• 14.95 miles

Area of a Circle & Semicircle

• Area of a full circle:

   $\pi r^2$  

• Area of a semi-circle:  $\frac{1}{2}\pi r^2$  

Area of a Quarter-Circle

• Area of a Quarter-circle =

   $\frac{1}{4}\pi r^2$  

If I want to find the area of an part of a circle, how would I do it?

Area of a Sector

• A sector is a fractional part of a circle

• To find the area of a sector:  $\left(\frac{n}{360}\right)\pi r^2$  

•  $n$  = degrees of the central angle of the sector.

•  $r$  = radius

Ex 1:

• Find the area of a sector with a central angle of

   $40\degree$  and a radius of 12 cm.

• Area of a Sector =  $\frac{n}{360}\pi r^2$  

•  $n=40\ \&\ r=12$  

• AoS =  $\frac{40}{360}\cdot\pi\cdot12^2$  

• AoS = $\frac{144}{9}\pi$  or  $50.27\ cm^2$  

Ex. 2:

• A pizza has a diameter of 12 inches and is cut into 10 equal slices. Find the area of 4 pizza slices, rounded to the nearest hundredth.

•  $n=\left(\frac{360}{10}\right)=36\degree$  &  $r=\left(\frac{12}{2}\right)=6$  

• Area of 1 slice:  $\frac{36}{360}\cdot\pi\cdot6^2$  

•  $=11.309733$  

• Area of  4 slices:  $4\left(11.309733\right)=45.24\ in^2$  

Multiple Choice

Find the area of a sector with a radius of 9ft and an angle measure of

 $132\degree$  

•  $10.37\ ft^2$  

•  $93.31\ ft^2$  

•  $29.7\ ft^2$  

•  $85\ ft^2$  

Multiple Choice

Find the central angle of the sector of the circle with a radius of 12 ft, and a sector area of 72 square feet.

•  $12.75\degree$  

•  $45\degree$  

•  $57.30\degree$  

•  $48.82\degree$