# Lesson 13

### 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.

### 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 $\ell=\theta \boldcdot r$.

Building Towards

### Glossary Entries

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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