Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
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.
Launch
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Activity
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Student Task Statement
Hexagon BCDEFG inside of circle with center A, radius = 1. Shaded region outside of hexagon. Distance between vertices of hexagon = 1. Perpendicular from A to hexagon side = the fraction square root of 3 over 2.
Calculate the area of the shaded regions.
Activity Synthesis
The purpose of this discussion is to ensure students understand that for a polygon inscribed in a circle, the more sides it is, the closer its area is to that of the circle.
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.
Display the images and ask students, “What’s the same? What’s different?” (Both are approaching the circle. One estimate is too small, and one is too large.)
Display the applet for all to see. Demonstrate the calculations for throughout the discussion.
Tell students that in mathematical language, we say that the perimeter of a polygon inscribed in a circle estimates the circumference. It gives us the lower bound for the circumference 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 us the upper bound.
Show students the formula for the perimeter of the polygon with the circle inscribed inside it is .
"What is the expression to approximate ?” ()
“What is the range for the value of starting with ?” ()
“What about ?” ()
Tell students that Archimedes, a Greek mathematician, did this without a calculator to evaluate the values of sine and 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 , which is impressively accurate for 250 BCE. Chinese mathematicians Liu (LEH-oh) and Chongzhi (chung-ZEE) 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 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 to still greater accuracy. In 2019, a team led by Emma Haruka Iwao (ha-ROO-ka ee-WAH-oh), a Japanese computer scientist, set the world record (at the time) by calculating over 31 trillion digits of .
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 information about circles. For example, we know how to calculate the area of regular polygons inscribed in a circle of radius 1.
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 and . To find , use the fact that a full rotation is and that in an isosceles triangle the altitude is also an angle bisector. So . , so is about 0.59 unit. so , is about 0.81 units. The area of the isosceles triangle is about 0.48 square unit 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. Mathematicians have been using this method to calculate the value of since at least 250 BCE, and they’re still working on it. In 2019, a team led by Emma Haruka Iwao (ha-ROO-ka ee-WAH-oh), a Japanese computer scientist, set the world record (at the time) by calculating over 31 trillion digits of .
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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.
Launch
Encourage students to refer to the examples from the previous lesson as they work to generalize.
Select students with different strategies, such as those described in the Activity Narrative, to share later.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For instance, review a concrete example, and keep it visible as students continue to work. Supports accessibility for: Organization, Attention
Activity
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Student Task Statement
Here is one part of a regular -sided polygon inscribed in a circle of radius 1.
Write a general formula for the perimeter of the polygon in terms of . Explain or show your reasoning.
Activity Synthesis
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?”
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.
Launch
Invite students to use the formula from the previous activity to calculate the perimeter of a square. (5.657 units) 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?” () “How close is the approximation?” ()
“Since the circumference is , we could use this formula to approximate . This is what mathematicians did before they knew the value of . Rewrite the formula to find an expression that gives the value of rather than .” ()
“How could we get a better approximation of than the square gives?” (More sides!)
Activity
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Student Task Statement
Let's use the expression you came up with to approximate the value of .
How close is the approximation when ?
How close is the approximation when ?
How close is the approximation when ?
How close is the approximation when ?
What value of approximates the value of to the thousandths place?
Student Response
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Building on Student Thinking
Activity Synthesis
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 .
MLR8 Discussion Supports. Display sentence frames to support students in producing statements to critique the reasoning of others. Examples:
“That could (or couldn’t) be true because .”
“We can agree that .”
“ and are different because .”
“Another way to look at it is .”
Advances: Speaking, Conversing
Standards Alignment
Building On
Addressing
HSG-GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
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.
