Lesson 11

Approximating Pi

11.1: More Sides (10 minutes)

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.

Square BCDE inside of circle with center A, radius = 1. Shaded region outside of square. Distance between vertices of square = the square root of 2.

Hexagon BCDEFG inside of circle with center A, radius = 1. Shaded region outside of hexagon. Distance between vertices of hexagon = 1.

Calculate the area of the shaded regions.

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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.

11.2: N Sides (15 minutes)

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.

Digital

Encourage students to refer to the examples from the previous lesson as they work to generalize.

Action and Expression: Internalize Executive Functions. Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.
Supports accessibility for: Organization; Attention

The applet shows a regular \(n\)-sided polygon inscribed in a circle.

Come up with a general formula for the perimeter of the polygon in terms of \(n\). Explain or show your reasoning.

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Encourage students to refer to the examples from the previous lesson as they work to generalize.

Here is one part of a regular \(n\)-sided polygon inscribed in a circle of radius 1.

Come up with a general formula for the perimeter of the polygon in terms of \(n\). Explain or show your reasoning.

Circle with radius = 1 unit. Isosceles triangle drawn inside of circle, left and right side lengths of triangle = radius of circle. One vertex of triangle at center of circle.

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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.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing their general formula for the perimeter of the polygon in terms of \(n\) and their reasoning. Allow students time to quietly circulate and analyze the formulas and reasoning in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight what information each group used, any calculations they completed, and how they expressed those procedures in a generalized formula.
Design Principle(s): Optimize output (for generalization); Cultivate conversation

11.3: So Many Sides (15 minutes)

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!)

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. To support student conversation, display sentence frames such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Let's use the expression you came up with to approximate the value of \(\pi\).

How close is the approximation when \(n=6\)?
How close is the approximation when \(n=10\)?
How close is the approximation when \(n=20\)?
How close is the approximation when \(n=50\)?
What value of \(n\) approximates the value of \(\pi\) to the thousandths place?

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Are you ready for more?

Describe how to find the area of a regular \(n\)-gon with side length \(s\). Then write an expression that will give the area.

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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 \(\pi\).

Speaking: MLR8 Discussion Supports. Use this routine to support students in producing statements to critique the reasoning of others when sharing their responses to the last question. Provide sentence frames for students to use, such as “That could (or couldn’t) be true because . . .”, “We can agree that . . .”, “_____ and _____ are different because . . .”, and “Another way to look at it is . . . .” Encourage students to press each other for details as they explain by requesting each other to elaborate on an idea or give an example.
Design Principle(s): Support sense-making

Lesson Synthesis

Display the images and ask students, “What’s the same? What’s different?” (Both are approaching the circle. One estimate is too small, one is too large.)

Display the applet for all to see. Demonstrate the calculations for \(\pi\) throughout the discussion.

“In mathematical language we say that the perimeter of a polygon inscribed in a circle estimates the circumference. It gives you the lower bound for the circumference of the circle since the perimeter is smaller than the circumference. Similarly, we say that the circumference of a circle inscribed in a polygon is estimated by the perimeter of the polygon, but since it is now outside the circle, the perimeter of the polygon gives you the upper bound. The formula for the perimeter of the polygon with the circle inscribed inside it is \(P=2n \boldcdot \tan \left( \frac{360}{2n} \right)\) What is the expression to approximate \(\pi\)?” (\(n \boldcdot \tan \left( \frac{360}{2n} \right)\)) “What is the range for the value of \(\pi\) starting with \(n=10\)?” (\(3.090<\pi<3.249\)) “What about \(n=50\)?” (\(3.140<\pi<3.146\))

“Archimedes did this without a calculator to tell him the values of sine or tangent. In fact he didn’t even have a concept of decimals! He was able to calculate the perimeter of a 96 sided regular polygon both inscribed in a circle and with a circle inscribed in it to say that \(3 \frac{10}{71} < \pi < 3 \frac{1}{7}\) which is impressively accurate for 250 BCE. Chinese mathematicians Liu and Chongzhi took a similar approach but found a method that was much faster to calculate and by 480 CE calculated the range of a 12,288-sided polygon which is accurate for the first eight digits. This was the most accurate approximation of \(\pi\) anyone could come up with for the next 800 years. Mathematicians from many other countries continued to independently discover and refine methods, and even today people are working on better ways to use supercomputers to calculate \(\pi\) to still greater accuracy. In 2019, a team led by Emma Haruka Iwao set the world record by calculating over 31 trillion digits of \(\pi\).”

Student Lesson Summary

It's easier to work with polygons than with circles because we can decompose polygons into simple shapes such as triangles. We can use polygons to figure out things about circles. For example, we know how to calculate the area of regular polygons inscribed in a circle of radius 1.

Pentagon inscribed in circle. 2 segments, length 1, from center to vertices form triangle. Dotted line, h, from center to triangle base forms 2 right triangles. Other right triangle leg is x. Angle between h and 1 is theta.

To find the area of this regular pentagon, let's find the area of one triangle and then multiply by 5. Drawing in the altitude creates a right triangle, so we can use trigonometry to calculate the lengths of both \(x\) and \(h\). To find \(\theta\) use the fact that a full rotation is \(360^\circ\) and that in an isosceles triangle the altitude is also an angle bisector. So \(\theta=360 \div 10\). \(\sin(36)=\frac{x}{1}\) so \(x\) is about 0.59 units. \(\cos(36)=\frac{h}{1}\) so \(h\) is about 0.81 units. The area of the isosceles triangle is about 0.48 square units and the area of the pentagon is 5 times that, or about 2.4 square units.

That's not very close to the area of the circle, but if we add more and more sides to the regular polygon, its area gets closer and closer to covering the entire circle.

Lesson 11

11.1: More Sides (10 minutes)

Warm-up

Student Facing

Student Response

Anticipated Misconceptions

Activity Synthesis

11.2: N Sides (15 minutes)

Activity

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Anticipated Misconceptions

Activity Synthesis

11.3: So Many Sides (15 minutes)

Activity

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Activity Synthesis

Lesson Synthesis

Lesson Synthesis

Student Lesson Summary

Student Facing