Deriving Formulas for Sector Area and Arc Length

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.

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.

While the formula for finding sector areas is fairly simple, the problem students will be doing in this section will provide plenty of challenge. I've found that this is a very good problem to make sure students really understand and are able to apply the formula.

Students will be working on the Complex Sector Area Problem (Solution on page 2). The problem requires some thought and students will definitely need to practice MP1: Making Sense of Problems and Persevering in Solving Them.

See the following screencast for an explanation of the solution.