Lesson 13
Using Radians
Lesson Narrative
In this lesson, students find sector areas and arc lengths for central angles with radian measure. They justify the formula for the area of a sector, and they observe that radian measure simplifies arc length calculations. As students explain why the expression \(\frac12 r^2 \theta\) gives the area of a sector with radius \(r\) units and central angle \(\theta\) radians, they are reasoning abstractly and quantitatively (MP2).
Learning Goals
Teacher Facing
 Interpret radian measure as the constant of proportionality between an arc length and a radius.
 Justify (in written language) why the formula $\frac12 r^2 \theta$ gives the area of a sector with central angle $\theta$ radians and radius $r$ units.
Student Facing
 Let’s see how radians can help us calculate sector areas and arc lengths.
Learning Targets
Student Facing
 I can calculate the area of a sector whose central angle measure is given in radians.
 I know that the radian measure of an angle can be thought of as the slope of the line $\ell=\theta \boldcdot r$.
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.