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

Fill in the Blanks

What is the measure of the central angle in a circle that intercepts an arc that is 18 cm in length and the radius of the circle is 7cm? Round the the nearest whole number. Numeric value only!

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

 $132\degree$  

•  $10.37\ ft^2$  

•  $93.31\ ft^2$  

•  $29.7\ ft^2$  

•  $85\ ft^2$  

Fill in the Blanks

Find the radius of a circle that has the sector area

 $68\ m^2$  and a central angle measure of  $63\degree$ . Round the the nearest hundredth.

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.

•  $12.75\degree$  

•  $45\degree$  

•  $57.30\degree$  

•  $48.82\degree$  

Fill in the Blanks

Find the area of the sector shown. Round to the nearest hundredth