# Deriving Formulas for Sector Area and Arc Length

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## Objective

SWBAT derive the formulas for the area of a sector and the length of an arc and explain why they make sense.

#### Big Idea

We're PROs at PORTIONS of circles.

## Activating Prior Knowledge

25 minutes

The essential understanding for this lesson is that proportional reasoning can be used to determine the lengths of arcs and the areas of sectors. To warm up to this idea (and at the same time work some Statistics into the mix) I start students off with some work on creating pie charts. Each student will receive APK_Deriving Sector Area Formula. I will have them work in groups of 2-4 to create pie charts using compass and protractor.

When students are done, I'll call students up to present their work. As they present, I'll be sure to emphasize our use of proportional reasoning.

## Deriving the Formulas

25 minutes

The formulas students will be deriving in this lesson are basically designed to calculate portions of circle area and circumferences. I start this section by giving students some specific cases where the portion of the circle is pretty obvious (e.g., a 90 degree arc represents 1/4 of a circle.). There are four such cases on the Deriving Formulas for Sector Areas and Arc Lengths resource. Each case is slightly less obvious than the ones preceding.

I have students complete the first four cases and then I call non-volunteers to come and present their solutions under the document camera. The main thing I emphasize is how students are determining the fraction of the circle that is represented by the arc or central angle measure.

After that, students will work on the general case. Then I'll call on non-volunteers again to make sure that we get the correct ideas out to all students.

Finally, students will work independently on the Reflection section of the handout to make sure that each student has understood how we derived the formulas.