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.
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.
- 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.
- 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.
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:
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
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
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:
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:
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.)
- The work performed and discoveries from the inscribed areas approach.
- The work performed and discoveries from the circumscribed areas approach.
- 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:
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:
- 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.
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.
- 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.]
- 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
- Were the students actively engaged and excited during the
lesson? If not, what could be changed to maintain higher levels of