Illuminations: Pi Filling, Archimedes Style!

# Pi Filling, Archimedes Style!

## Inscribed and Circumscribed Polygons

 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.

### 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.

### Materials

 Inscribed and Circumscribed Activity Sheet Scientific or Graphing Calculator

### Questions for Students

 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.] 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.] 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.]

### Assessment Options

 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. 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. 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: 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. The 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. The 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: Hit PRGM, right arrow over to NEW, hit ENTER. Name the program (max 8 characters), hit ENTER. The colons start every line, and do not need to be typed in; hit ENTER to start the next line. The Prompt and Disp commands can be found under PRGM, then right arrow to I/O. When finished, hit 2nd-QUIT. 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.

### 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?

### NCTM Standards and Expectations

 Algebra 9-12Generalize patterns using explicitly defined and recursively defined functions. Geometry 9-12Use trigonometric relationships to determine lengths and angle measures.

### References

 Slowbe, Jason. 2007. Activities for Students: Pi Filling, Archimedes Style! Mathematics Teacher 100: 485.
 This lesson prepared by Jason Slowbe.

1 period

### NCTM Resources

 Principles and Standards for School Mathematics (Book and E-Standards CD)

### Activities

 More and Better Mathematics for All Students
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