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Inscribed and Circumscribed Polygons

  • Lesson
9-12
1
AlgebraGeometry
Jason Slowbe
Location: unknown

By calculating the areas of regular polygons inscribed and circumscribed about a unit circle, students create an algorithm that generates the never-ending digits of π, a common curiosity among high school students.

The purpose of this lesson is to use the geometry of regular polygons inscribed in and circumscribed about the unit circle to create an algorithm for generating the digits of π, known to be the area of the unit circle. The mathematical results are motivated by numeric examples, enabling students to investigate patterns in these numeric examples and to extend the results to the general case. Ultimately, students will investigate the inscribed and circumscribed methods simultaneously with a partner, and will determine which method provides a better approximation for the numerical value of π.

Students should consult with a designated partner when they have questions, get stuck, or need to check their results, but each student should do the mathematics and complete the Activity Sheet individually, unless otherwise noted. Each student will need a scientific or graphing calculator, an Activity Sheet, and scrap paper.

Students should already be familiar with the formula for finding the area of a regular polygon, A = ½(ap), where a is the apothem and p is the perimeter, and with the sine, cosine, and tangent functions before starting this lesson.

To begin the lesson, distribute the Inscribed and Circumscribed activity sheet and direct students to the four diagrams of a regular triangle, square, pentagon, and hexagon inscribed in unit circles. Briefly explain to students that Archimedes analyzed the geometry of regular polygons inscribed in unit circles, then note that the numbers of sides of the regular polygons starts with 3 (triangle) and increases by 1 side in each diagram. Ask students to predict what would happen when the number of sides of the regular polygons approaches infinity, and then have a few students share their predictions, as well as any other thoughts or questions that come to mind, with the class.

pdficon  Inscribed and Circumscribed Activity Sheet 

Though students will most likely have difficulty understanding the concept of a polygon with an infinite number of sides, asking them to think about this phenomenon will guide their thinking for the remainder of the lesson. Instruct students to keep these diagrams in mind throughout the lesson, because they will be asked to describe mathematically what happens as the number of sides in the regular polygons increases to infinity.

Actively guide students through the process of finding the area of the regular triangle first, following the steps below, as these steps will be replicated by students for other polygons. Ask leading questions that clearly communicate the connections between the prior step and the goal of the next step. A sample question of this form is: "We now know the base angles measure 30°. How does knowing this angle measure help us find the length of the apothem?"

Explain that the goal is to find the area of a regular triangle inscribed in a unit circle.

2477 equi tri
  1. Consider the inner isosceles triangle below, formed by drawing two radii. Note that each base angle was 60°, from (n – 2) × 180/n, and the radii divide the base angles in half to form 30° angles.
2477 inner isos
  1. Next, form a right triangle by drawing the apothem (an altitude) from the center of the circle perpendicular to the base. Using the sine and cosine functions, the lengths of the apothem, y, and the base, 2x, can be found. Be sure to use the "degrees" setting on your calculator.
2477 inner isos labeled
2477 eqn 1        2477 eqn 2

Consequently, the length of the base is 2x = 2 cos 30°, and the apothem is y = sin 30°.

We now have enough information to calculate the area of the inscribed regular triangle:

2477 eqn 3

It is these last two lines, an exact expression for the area and its decimal approximation rounded to the ten-thousandths place value, that should be transferred to the table on the activity sheet. Throughout, it is very helpful to organize the calculations as written above, so students can see how the angle measure, perimeter, and apothem were calculated.

Working with their partners, instruct students to draw a picture of the regular triangle circumscribed about the unit circle, and then to work together with their partners in finding its area, following the same format as the inscribed triangle’s calculations. Circulate to each group, ensuring that they are on the right track. If students struggle to understand particular steps in finding the area, or are getting lost in all of the calculations, refer students to the picture of the circumscribed triangle. Reminding students of the geometric representation for their symbolic calculations can be an effective way to prevent confusion. The area of the circumscribed triangle is 3 ÷ tan 30° ≈ 5.1962, and should be transferred into the table on students’ activity sheets.

With students working together, direct the students to find the inscribed and circumscribed areas of the square. Check each group’s work to make sure they get an inscribed area of 2.0000 and 4.0000, respectively. Once each group has obtained the correct areas for the squares, have one student in each group find the area of the inscribed pentagon and circumscribed hexagon, with the other student finding the area of the circumscribed pentagon and inscribed hexagon. When each student has completed these two calculations, have each student analyze their partner’s calculations for correctness, discussing and debating each others’ work until both students agree that all four areas are correct.

Once students have completed the inscribed and circumscribed areas through the hexagon, ask students to look for patterns in their work, with the goal of generalizing an area function for the inscribed areas in terms of n. Watch for students who use an incorrect expression for the angle measures, as this is a common error. If students do not yet see the pattern, instruct them to calculate the area of the inscribed 7‑gon, and beyond if necessary, until this pattern becomes clear:

2477 eqn4

Watch for students who are unable to generalize to this function. Point out the similarities in their work, emphasizing the calculation of the perimeter and the apothem. Once the students obtain this function, they should use it to complete the rest of the table, including n = 50 and n = 200, and then should discuss Question 2 with their partner.

Teaching Tip: Watch for students who wonder what happens when this area function takes the value n = ∞. They may reason that when n = ∞, this area function reduces to n · cos 90° · sin 90° = n · 0 · 1, which has a value of 0. Students may wonder how the areas of the n‑gons increase, but when they reach infinity, the areas jump back to zero? The best answer is that ∞ is a strange concept, especially before students have taken calculus. Technically, the area function would reduce to ∞ · 0 · 1, which is indeterminate. Other indeterminate forms include 0/0, ∞/∞, ∞ · 0, 00, ∞0, and 1. One last curious result involving infinity would be to ask students to evaluate the fraction sin θ/θ as θ approaches 0. Revealing that the limit equals 1 is usually enough for students to accept that infinity doesn’t work like an ordinary "number."

Likewise, ask students to analyze the patterns in the circumscribed polygons’ areas, in order to define a general area function in terms of n. Students should discover this area function:

2477 eqn 5

Students will complete this part of the lesson at different times, but all should be far enough along such that holding a class discussion will make sense to everyone. As students finish, point out that the circumscribed areas approach π just like the inscribed areas did, except they are always greater than π. Refer to the diagrams at the top the Activity Sheet to justify why the inscribed areas will always be less than the full area of the unit circle, and why the circumscribed areas will always be greater than the area of the unit circle. Emphasize that increasing n causes the regular polygons to become more "circle-like," which is why their areas approach the area of the unit circle, known to be π. An important question to ask students in this discussion is whether the areas will ever reach π. Drawing the distinction between the area being exactly equal to π, and the area matching the digits of π to a certain place value, is a common misunderstanding from students. Mention to students that this notion of a limit is the fundamental idea for studying calculus.

Teaching Tip: As an alternative method for generating results, you can have students use the table features on their graphing calculators. Students can generate the area of inscribed polygons in the first list and the area of the circumscribed polygons in the second list, and they can generate the perimeter of inscribed and circumscribed polygons in the third and fourth lists. Students should notice that the first and third lists approach π from below and the second and fourth lists approach π from above.

The final component of this lesson is to ask students which method they prefer for generating the digits of π. Emphasize that both methods, if continued for large enough values of n, will get arbitrarily close to the true value of π, but ask whether one of the methods is preferable. Valid responses include, but are not limited to:

  • For each particular n (except n = 3), the circumscribed area is always closer to π than the corresponding inscribed area.
  • Visually, the circumscribed polygons conform more closely to the circle. Another way to think about this concept is to look at the leftover areas compared to the unit circle. There is less area outside the circle and inside the circumscribed polygon; likewise, there is more area inside the circle and outside the inscribed polygon.

Students should be able to explain that the method of approximating π from circumscribed regular polygons is more efficient than from inscribed regular polygons.

For closure, select three students, one each to explain the following in their own words. (You can pretend that each student is explaining to a hypothetical student who was absent for this lesson.)

  1. The work performed and discoveries from the inscribed areas approach.
  2. The work performed and discoveries from the circumscribed areas approach.
  3. How the inscribed and circumscribed areas are similar and different, and how both methods relate to approximating π.

Remaining class time can be spent with some of the assessment questions if only a few minutes remain, or pursuing one of the extensions if a significant amount of time remains.

Selected Solutions to the Activity Sheet 

Question 1. The area formula for the inscribed regular n-gon is:

2477 eqn 7

Question 2. For n = 3, the area is 1.2990. For n = 200, the area is 3.1411. As the value of n continues to increase, the area of the inscribed polygon will approach the area of the unit circle, which is π.

Question 4. Circumscribing regular polygons around the unit circle will overestimate the value of π, and therefore using these methods with circumscribed polygons will generate areas that approach π from above. The area formula is:

2477 eqn 8

Assessments 

  1. Consider the regular triangle inscribed in a circle with r = 2 and A = 3√3. Find the perimeter of the triangle. [6√3.] This question assesses whether students can use the proper trigonometry functions to find the apothem, and then use the formula A = ½(ap) to solve for p.
  2. As the number of sides n of regular polygons inscribed in the unit circle increases, will the areas ever reach π? [No. The regular polygons fill more and more of the area of the unit circle, but never become an exact circle.] This question assesses whether students understand the nature of the approximations to π found in the lesson.

        Refer to the regular triangle circumscribed about a circle, shown below.

2477 circ equi tri

Modify your work from earlier in the lesson to calculate how the area of δABC would change if the     radius of the circle were 2. [The area is quadrupled.] This question assesses whether students can analyze and modify their previous work from the lesson.

Extensions 

This lesson involves repeating the same calculations with different values of n. The programming features on the graphing calculator can automate this procedure, giving students more time to explore, make conjectures, and interpret their mathematical results. No background in programming is required for this extension.

First, direct students to the explicitly defined function for the area of the inscribed regular n-gons, and remind students that this function depends only on n; that is, the area of any particular inscribed n-gon can be found knowing nothing more than the number of sides n. Students discovered in the lesson that the general function for the area of an inscribed n-gon is:

2477 eqn6

This expression can be simplified; however, it may help students to leave it in this form so they can see precisely where the perimeter, apothem, and angle measure calculations came from.

2477 calc commandsThe first step is to store the desired value for n into the calculator. This is done by entering the desired value for n, then pressing the STO→ key directly above the ON button, and then ALPHA‑N. Pressing ENTER stores the desired value into the calculator’s memory for that particular variable, and anytime that variable is used in an expression, the calculator will substitute the desired value into the expression. (See figure). Then, typing the expression above gives the area of the desired n-gon. To calculate the area for a different value of n, repeat the steps above to store the new value for N, and then type 2nd-ENTRY (twice) to recall the long area expression so students do not have to retype the entire expression.

Even this method of restoring new values of n and recalling the expression for area is a bit tedious. Writing these same operations into a program removes much of the hassle.

2477 programThe code for such a program can be entered as shown. Alternatively, download the InscPoly.8xp program by clicking on the link, choosing "Save Target As…," and transferring the program to your TI‑83 or TI‑84 graphing calculator.

To then use the program, follow these steps:

  1. Hit PRGM, right arrow over to NEW, hit ENTER.
  2. Name the program (max 8 characters), hit ENTER.
  3. The colons start every line, and do not need to be typed in; hit ENTER to start the next line.
  4. The Prompt and Disp commands can be found under PRGM, then right arrow to I/O.
  5. When finished, hit 2nd-QUIT.
  6. To run the program, hit PRGM, then select the program from the list that appears.
Command Locations
  • Degree – in MODE
  • Disp, Prompt, ClrHome – PRGM, then I/O
  • Lbl, Pause, Goto – PRGM

Note that all commands to be used in the program can also be found in the Catalog. Enter the Catalog by pressing 2nd‑0. This will take you to a list of all possible commands for use in programming. To jump to a different location in the Catalog, use the Alpha keys to enter a letter, and you will be forwarded to the first item in the Catalog that begins with that letter. For instance, if you enter P, you will be taken to the command "Param."

Some students really like to learn about programming, which can be a great way to engage and challenge those students in other lessons throughout the year.

Questions for Students 

1. Will the inscribed areas ever reach pi?
[No, because the straight sides of polygons will never curve the same way as a circle, but the areas can get as close as we want with a large enough number of sides.]
2. For the regular polygons inscribed in the unit circle, what is the range of values that the apothem can have? The perimeter? How do the apothems and perimeters change relative to each other? Explain your reasoning.
[The apothem is at its shortest length with the regular triangle and increases as the number of sides in the polygon increases. As the number of sides increases, the apothem gets closer and closer to the radius of the circle. As the apothem approaches 1, the perimeter of the polygon approaches 2π, the circumference of the circle.]
3. For any particular n-gon about the unit circle, would you expect the inscribed or the circumscribed n-gon to give a closer estimate of pi? Explain your reasoning.
[The circumscribed n-gons. There is less "wasted area" between the circle and the n-gons.]

Teacher Reflection 

  • What modifications did you make to ensure that students of various ability levels were engaged in the lesson?
  • What feedback did you receive from students to indicate that enough numeric examples were provided to offer an appropriate amount of guidance for making generalizations?
  • What questioning strategies were effective in stimulating students to interpret results and make predictions for larger values of n?
  • Did the numeric examples provide enough guidance for students to generalize their results to larger n-gons?
  • Did students have a clear understanding of the end goal or purpose for this lesson as they were working? Were you able to effectively stimulate the students to interpret their results and make predictions for what should happen for larger values of n?
  • Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
  • Were the students actively engaged and excited during the lesson? If not, what could be changed to maintain higher levels of student participation?
 
Geometry

Improving Archimedes' Method

9-12
Archimedes was the first mathematician to develop a converging series approximation to π. That highly influential discovery guided the development of calculus many hundreds of years later. However, his method only gives lower and upper boundaries that form intervals known to capture π, not a single numeric estimate of π. In this lesson, students ask, “Where is π located in those intervals?” They also discover an improvement to Archimedes' method that generates the infinite digits of π more efficiently and accurately.

Learning Objectives

By the end of this lesson, students will:

  • Calculate the areas of regular polygons using the formula ½(ap).
  • Write explicit functions for the areas of inscribed and circumscribed regular n-gons.
  • Use trigonometric functions to find side lengths of triangles.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.
  • CCSS.Math.Practice.MP8
    Look for and express regularity in repeated reasoning.