Understand Radian Measure

16.2 Arc Length and Radian Measure

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

Arc Length (COPY NOTES)

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.

(NOTES)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 

(NOTES) 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.

Fill in the Blanks

Multiple Choice

Find the arc length of the bold arc.  Leave π  in your answer.

$$S=\frac{\theta2\pi r}{360\degree}$$  leave π in your answer.

• 5688π ft

• 15.8π ft

• 150.4π ft

• 54,150 ft

Multiple Choice

Find the arc length. Leave your answer in terms of π.

$$S=\frac{\theta2\pi r}{360\degree}\ leave\ the\ \pi\ in\ the\ answer.$$  

• 7π/4

• 7π/2

• π/8

• 45π

Multiple Choice

When she is outdoors, Tasha, the dog, is tied to a stake in the center of a circular area of radius 24 feet. The angle between her dog house and her favorite hydrant is 165 degrees. What is the length of the arc from her dog house to the hydrant, following minor arc DG, to the nearest foot.

• 40

• 35

• 55

• 69

Multiple Choice

An ____ is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.

• adjacent

• arc

• algorithm

• area

Multiple Choice

What is the formula for?

• arc length

• area of a sector

• area of a segment

• arc measure

Multiple Choice

Find the arc length of the bolded arc in the picture. (Use 3.14 for $$\pi$$  )

$$S=\frac{\theta\cdot2\cdot\pi\cdot r}{360}$$  

• A

• B

• C

• D

Multiple Choice

In circle O, the radius is 4, and the length of minor arc AB is 4.2 feet. Find the measure of minor arc AB to the nearest degree. (Use 3.14 for $$\pi$$  )

• 90

• 60

• 55

• 113

Multiple Choice

What formula is this?

• Arc length when θ is in radians

• Arc length when θ is in degrees

• Sector Area when θ is in radians

• Sector Area when θ is in degrees

Multiple Choice

A central angle measures 4.5 radians and has an arc of 35 inches. What is the radius of the circle?

• 7.78 inches

• 8.21 inches

• 157.5 inches

• 7 inches

Multiple Choice

Find the arc length.

• 7 cm

• 28π cm

• 7π cm

• 28 cm

Multiple Choice

• 12π cm

• 27π cm

• 12 cm

• 27 cm

Multiple Choice

Find the arc length.

• 3π/4 ft

• 39π/4 ft

• 13π/4 ft

• 4π/13 ft

Multiple Choice

What is 225 degrees in radians?

• 7π/4

• 5π/4

• 3π/4

• 5π/3

Multiple Choice

Convert 150⁰ to radians.

• $$\frac{5\pi}{6}$$

• $$\frac{3\pi}{4}$$

• $$\frac{7\pi}{6}$$

• $$\frac{2\pi}{3}$$

Multiple Choice

What is 7π/4 radians in degrees?

MULTIPLY BY (180/π)

• 300

• 135

• 315

• 45

Multiple Choice

 Convert  $$\frac{\pi}{4}\ radians$$  to degrees

• 90⁰

• 45⁰

• 60⁰

• 30⁰