The goal of this activity is to further reinforce the concept that polygons with many sides are nearly circular. Students find the difference in area between a square and the circle it is inscribed in, then compare it to the difference in area between a hexagon and the circle it is inscribed in. It is also an opportunity to practice decomposing a shape, which will be essential to the generalization in this lesson.

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.

In this activity students will use the formula they developed in the previous activity. They will see how quickly this formula approximates  and consider how accurate the approximation is for polygons of various side lengths.