Lesson 11
A New Way to Measure Angles
Lesson Narrative
In this lesson, students prove that the length of the arc intercepted by a central angle is proportional to the radius of the circle. Then, they learn that the ratio of arc length to radius is called the radian measure of an angle. They use string to measure arc length in terms of a circle’s radius, and connect the results to the definition of radian angle measurement.
An understanding of radian measure will be necessary in future courses when students extend the domain of trigonometric functions using the unit circle.
Students have an opportunity to reason abstractly (MP2) as they consider the connection between radian measurements and arc lengths.
Learning Goals
Teacher Facing
 Comprehend (in spoken and written language) that the radian measure of an angle whose vertex is the center of a circle is the ratio of the length of the arc defined by the angle to the circle’s radius.
Student Facing
 Let’s look at a new way to measure angles.
Required Preparation
Be prepared to display an applet for all to see during the lesson synthesis.
Learning Targets
Student Facing
 I know that the radian measure of an angle whose vertex is the center of a circle is the ratio of the length of the arc defined by the angle to the circle’s radius.
CCSS Standards
Glossary Entries

radian
The radian measure of an angle whose vertex is at the center of a circle is the ratio between the length of the arc defined by the angle and the radius of the circle.

sector
The region inside a circle lying between two radii of the circle.
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