If students are struggling, ask them what shapes they could decompose the hexagon into. (two trapezoids or six triangles)

In this activity students build off the specific calculations from the previous lesson to generalize the perimeter of a polygon inscribed in a circle of radius 1. The relatively unstructured presentation of this activity is purposeful (MP1). Students work with their groups to determine what information they need, how to calculate in the specific cases, and how they can express those repeated procedures in a generalized formula (MP8).

Monitor for groups who have a clear representation of one or more aspects of the process. Here are some generalizations students might make, ordered by how students are likely to work through the generalizing process to get to the final formula:

• Generalize the angle measure
• Generalize the segment length
• Extend to the whole perimeter

Look for groups who have an annotated diagram or a concrete example side by side with an expression using variables for each step and sequence them in the order of the calculation.

In the digital version of the activity, students use an applet to visualize inscribed and circumscribed regular polygons for a circle of radius 1. The applet allows students to quickly and accurately see how the values of the perimeter and area of the polygon changes as the number of sides change. The digital version may be helpful for students who benefit from dynamic visuals or for checking that generalized formulas are correct.

If students don't have individual access, displaying the applet for all to see would be helpful during the Launch.

The purpose of this discussion is to ensure students understand the components of the formula .

Invite previously selected students to share their representations. Sequence the discussion of the methods in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses by inviting another student to summarize by explaining where each piece of appears in the diagrams and concrete examples presented.

Connect the different responses to the learning goals by asking questions such as:

• “How many different polygons did you calculate the perimeter for before trying to work out the general formula?”
• “From the work just shared, did you see anyone organize their thinking in a way you found particularly helpful for seeing the general formula?”
• “Which was more challenging to determine, the general angle measure or the segment length?”

The purpose of this discussion is for students to consider why two values of n that approximate to the thousandths place may be correct. 

Invite students to share the values of they chose and how close to the approximation is. Invite students who chose 72 and students who chose 102 to debate. (72 sides is enough because there are 3 accurate digits after the decimal place. 72 sides isn't enough. We need 102 sides in order for the approximation rounded to the thousandths place to be correct.)

Share that people often employ this kind of thinking to program calculators to get very accurate approximations without the calculator needing to store a very long string of digits to represent .

If students are struggling, invite them to go back to the problems from the previous lesson to generalize the process. (Draw in the altitude. Find the measure of the central angle. Find the length of the opposite leg.) Suggest that students generalize each step before trying to write a single formula.