3.2 - Arc Length
Contents
Objectives:
- Understand arc length as a proportion of the circumference of a circle.
- Find the arc length given an angle on circles of varying radii.
- Find an angle given arc length and radius.
- Definitions
- arc, arc length
Lesson Guide
Arc Length Exploration
Based on their work in Section 15.1, students are hopefully comfortable with arcs as portions of a circle.
An arc is a portion of the circumference of a circle (sketch an example here to illustrate).
Have students complete Problem 1 on the worksheet, and discuss their strategies for finding the arc lengths as a class.
Radians and Arc Length on the Unit Circle
An important connection is that between the ideas of arc length and radians. Have students recall that there are $2\pi$ radians in a circle and the circumference of the unit circle is $2\pi$ units.
Radians correspond to arc length on the unit circle.
Give a couple of examples (i.e. half of the way around the unit circle is $\pi$ radians, and the length of the arc corresponding to traveling half of the way around the unit circle is $\pi$ units). It may be helpful to draw pictures here to make the distinction between arc lengths and angles. Make sure that your examples don't clash with Problem 1.
-Example 1:
-Example 2:
Have students work through Problem 2.
Arc Length on other Circles
Suggest that we would like to find the arc length defined by an angle in radians on any circle, and give an example or two to show how we might do this.
-Example 1: Find the arc length defined by $1/2$ radians on a circle of radius $3$.
-Example 2:
Conclude that
The length $s$ of an arc defined by angle of $\theta$ radians on a circle of radius $r$ is $s = r\cdot \theta$.
Have students complete the worksheet.
Comments
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