Lesson 11

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.)

Use the applet to demonstrate what happens to the areas of the polygons as the number of sides increases.

“What if the polygon has 10 sides? 20?” (The shaded region would be very small.) Reinforce the idea that the more sides an inscribed polygon has, the closer it is to a circle.

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. 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 generalized any part of the process.

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 students generalize each step before trying to write a single formula.

Invite previously selected groups to share one step at a time:

• generalize the angle measure
• generalize the segment length
• extend to the whole perimeter 

Invite another student to summarize by explaining where each piece of \(P=2n \boldcdot \sin \left( \frac{360}{2n} \right)\) appears in the diagram.

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

Invite students to use the formula from the previous activity to calculate the perimeter of a square. (5.657) Tell students to round to the thousandths place for this activity. “Does that seem close to the perimeter of the circle? What is the circumference of a circle with radius 1?” (\(2\pi=6.283\)) “How close is the approximation?” (\(6.283-5.657=0.626\))

“Since the circumference is \(2\pi\) we could use this formula to approximate pi. This is what mathematicians did before they knew the value of pi. Rewrite the formula to find an expression that gives the value of \(\pi\) rather than \(2\pi\).” (\(n \boldcdot \sin \left( \frac{360}{2n} \right)\))

“How could we get a better approximation of \(\pi\) than the square gives?” (More sides!)

Invite students to share the values of \(n\) they chose and how close to \(\pi\) 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, you need 102 sides in order for the approximation to round to the thousandths place correctly.)

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 \(\pi\).