Lesson 13

Using Radians

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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

Addressing

Building Towards

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.