Areas of Regular Polygons Inscribed in Circles

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SWBAT express the ratio of an inscribed regular polygon's area to the area of its circumscribed circle as a function of the number of sides.

Big Idea

Ever seen a regular polygonal eclipse? Tune in to this lesson to find out what you've been missing.

Making Preliminary Predictions

20 minutes

In this section, students will be getting familiar with the context of the lesson and making some preliminary predictions. Each student receives Inscribed Regular Polygon Area Ratios_Predictions.

Item 1 asks students to estimate the ratio of a regular heptagon to that of its circumscribed circle and express that ratio using only a single number. I've found that my students are sometimes not sure how to express a ratio as a single number so I explain how 3:2 can be expressed as 1.5, for example. Once students have made their estimates, I'll do a quick whip around to hear students estimates and how they arrived at them.

Item 2 requires students to do some visualization in order to understand the question. I have the students do a quick pair share to discuss what it means to vary the radius. Then I'll have them discuss their answers to the question. After that, I'll call on some non-volunteers to share what they've concluded.

Next we'll proceed to work through item 3 in the same way we did for item 2.


Item 4a asks students to identify the practical domain of a function. After students have had a chance to discuss their answers with their partners, I'll call on students randomly to share their answers. As they do I'll provide the necessary coaching to make the answers more precise. For example, if a student says, "All numbers from 3 to infinity...," I"ll ask, "Does that include 3? How about 5.7?"

Finally, it's time for students to sketch their conception of the function graph. I require that this be a totally independent task for each student. This way, each student will communicate his or her understanding, which will be important for them to reflect upon later in the lesson.

Group Activity

30 minutes

In this lesson, students will be working in groups of four. The goal is for students to come up with a general formula of the ratio of a regular polygon's area to the area of its circumscribed circle as a function of the number of sides the polygon has. In order to reach this goal, each student in a group of four will be closely working with one regular polygon (square, pentagon, hexagon or octagon) and then the group members will be coming together to abstract to the general case based on their findings in the specific cases.

Before the lesson, I'll photocopy the first four pages of Area Ratio of inscribed regular polygon to circumscribed circle, each with the fifth page (The General Case) photocopied on the back. This way each student in a group will have his or her specific case on one side and the general case on the back.

Before getting students into group (i.e., while they are still facing forward in rows) I'll want to do some basic orientation to the task. First I call attention to the Long Term Goal and have students discuss what they think it means. Then I'll have a couple of share outs, clarifying or elaborating as necessary to make sure students understand the goal.

Then I'll explain that we will be writing areas in terms of r since the area of a circle is already in terms of r. Students have had a good deal of practice writing areas of regular polygons in terms of different variables in the previous lesson so I feel comfortable turning them loose on this task.

I have the students get into groups of four and I hand each student in a group a different specific case. Then I direct the students to start working. I explain that they are all working on similar but slightly different problems so they will be able to get general guidance, but not specific answers, from their partners.

When students have had enough time to complete the work for their specific cases, I'll direct the groups to turn their focus to the back side which asks them to record the four specific cases and determine the rule for the general case. At this time, I'll walk around the classroom to see students having aha! moments or having questions. If groups finish early, I'll have them get to work on the Extension.

Finally, I'll demonstrate the process and rationale for re-writing the function using the double angle formula (See Ratio of Inscribed to Circumscribed_KEY)

Plenary Discussion

20 minutes

Now that the groups have finished their work, I have the students return the desks to row seating formation. Then I show and discuss the work for one of the specific cases and re-cap how all of the specific cases turned out and how we used those to abstract to the general case.

Then all that's left to do is to determine what happens with the function (i.e., the ratio between the area of a regular polygon and the area of its circumscribed circle) as n approaches infinity. I do that via a demonstration on DESMOS.

Check out the screencast to get an feel for the content and style of that demonstration.


After seeing this demonstration, we'll have some discussion about limits, asymptotic behavior, discrete vs. continuous functions, etc. Not that these are the learning goals of the lesson, but it doesn't hurt to expose students to important concepts and language at this early stage.

Reflection and Closure

10 minutes

Now that students have been through the lesson, it's time for them to reflect on what they learned and make any necessary revisions to their original thinking based on what they've learned. Page two of Inscribed Regular Polygon Area Ratios_Predictions, provides a space for students to do just that. So at this point, I give students 7-10 minutes to complete the Post-Activity Reflection and then prepare to turn in their work.