Arc Length

Arc Length

Objectives

• Recognize the formulas for finding the arc length given degrees or radians.

• Calculate a sector's arc length

Definition

• Arc Length: the section of the circumference of a circle between 2 radii.

• Arc length measured using radians.

Kinds of Arcs

• Minor Arc: Arc created by an angle of $180\degree$ /   $\pi$  radians or less 

• Major Arc: Arc created  by an angle or more than  $180\degree$  / $\pi$  radians

• Semicircular Arc:  Arc created when both angles are equal   

Formula for Arc Length

• If we begin with Degrees: $\frac{\pi}{180}r\theta$   

• If we begin with Radians  $r\theta$  

Example

Find the length of the blue arc given the radius of the circle is 8 cm, and the angle measure shown is in radians. Give the answer to one decimal place.

Multiple Choice

Find the arc length indicated by the bolded arc.

• 8.3 in

• 7.2 in

• 6.8 in

• 6.4 in

Multiple Choice

Find the arc length indicated by the bolded arc.

• 17.3 km

• 79.1 km

• 95.0 km

• 99.0 km

Multiple Choice

Find the arc length indicated by the bolded arc.

• 8.4 mi

• 9.4 mi

• 60.2 mi

• 117.8 mi

Example

A pendulum of length 26 cm swings  $58\degree$ .  Find the length of the circular pathway that the pendulum makes giving the answer in centimetres in terms of 𝜋. 

Multiple Choice

Find the arc length of the bolded arc.

• A

• B

• C

• D

Multiple Choice

Find the length of the bold arc. Leave  $\pi$   in your answer.

• 5688π ft

• 15.8π ft

• 150.4π ft

• 54,150 ft

Multiple Choice

The formula for the arc length given an angle measurement in degrees is

•  $\frac{\pi}{180}r\theta$  

•  $\frac{\pi}{360}r\theta$  

•  $\pi r^2$  

Multiple Choice

The formula for the arc length given an angle measurement in radians is

• $2\pi r\theta$

• $r^2$

• $r\theta$