Arcs

7.4 Arc Length

Goal: I can find the arc length, radius/diameter, or central angle of a circle.

Arc Length

Arc Length is the length of a portion of the circumference.

To find this we multiply the circumference by the fraction of the circle the arc subtends.

The fraction of the circle is determined by dividing the central angle by 360.

Example 1: Find length of minor arc AB.

• 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 

Example 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

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

•  $7=\frac{1}{5}\pi r$  

•  $r=11.140846$  

• The diameter of the pizza is 22 units.

Multiple Choice

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

• 12.04 cm

• 24.08 cm

• 144.51 cm

• 72.26 cm

Multiple Choice

What is the radius of a circle that has a central angle of  $$230\degree$$  and an arc length of 60 miles?

• 6.10 miles

• 29.89 miles

• 74.95 miles

• 14.95 miles

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