Intermath | Workshop Support
Write-up
Title
N-Sided Circle?
Problem Statement
Draw a circle of diameter 1. Using the polygon script function in GSP, inscribe an
equilateral triangle inside the circle. Measure its perimeter. Repeat this process with
polygons with an increasing number of sides. (There are scripts to inscribe a square, a
pentagon, an octagon, and a seventeen-gon.) What can you say about the perimeters of
these polygons when you compare them to the circumference of the circle? Repeat this
investigation with a circle of diameter different from 1.
Problem Setup
How does the perimeter of a regular polygon compare with the circumference of a circle?
As the polygon gains sides, it appears to have more rounded edges (larger interior
angles). As it approaches infinite sides, the perimeter approaches the circumference of a
circle.
Definitions
Chord: A line segment that joins two points on the circumference of circle
Circumference: The distance around a circle.
Inscribed Polygon: A polygon is inscribed in a circle if and only if each of its vertices lie
on the circle.
Perimeter: The sum of the lengths of the sides of a polygon.
Regular Polygon: A polygon that is equiangular (all angles are equal) and equilateral (all
sides are equal).
Circle: The set of all points in a plane that are the same distance, called the radius, from
a given point, called the center.
Polygon: A closed figure formed by three or more line segments.
Plans to Solve/Investigate the Problem
1. Use Geometer’s Sketchpad to create a circle.
2. Create a point on the circle and segment from the point to the center (radius).
3. Use transformations (Menu title) to create points an equal distance apart on the
circle. (A circle has 360 degrees.)
4. Measure the perimeters of the polygons and the circumference of the circle.
Investigation/Exploration of the Problem
I carried out the plans stated above. After creating the circle on Geometer’s Sketchpad, I
created the regular polygons beginning with an equilateral triangle. Using the
transformation menu and my previous knowledge that a circle has 360 degrees, I put
points on the circle an equal distance around the circumference to create the points for the
regular polygon.
Below are examples for selected regular polygons. The figure is small due to the original
problem that requested a diameter of 1.
Equilateral Triangle (Rotate points 120 degrees.)
Perimeter
Perimeter
ABC = 2.61 cm
ABC = 7.31 cm
B
m AB = 2.44 cm
m AB = 0.87 cm
m AC = 0.87 cm
A D
C
m CB = 0.87 cm
Circumference
DE = 3.15 cm
B
E
A
m AC = 2.44 cm
D
m CB = 2.44 cm
Circumference
E
DE = 8.84 cm
DC = 1.41 cm
DC = 0.50 cm
C
Square (Rotate points 90 degrees.)
Perimeter BACF = 2.82 cm
Perimeter BACF = 8.99 cm
m AB = 0.71 cm
m AB = 2.25 cm
m AC = 0.71 cm
B
A
F D E
C
m CF = 0.71 cm
m FB = 0.71 cm
Circumference
DE = 3.14 cm
m AC = 2.25 cm
A
m CF = 2.25 cm
D
F
m FB = 2.25 cm
Circumference
DC = 0.50 cm
B
E
DE = 9.99 cm
C
DC = 1.59 cm
Regular Pentagon (Rotate points 72 degrees.)
Perimeter BFGCA = 2.93 cm
Perimeter BFGC A = 9.19 cm
m FB = 0.59 cm
m FB = 1.84 cm
m AB = 0.59 cm
m AB = 1.84 cm
B
m CA = 0.59 cm
F
m GC = 0.59 cm
D A
G
CE
m FG = 0.59 cm
Circumference
B
F
m CA = 1.84 cm
D
m GC = 1.84 cm
DE = 3.14 cm
Circumference
DC = 0.50 cm
A
G
m FG = 1.84 cm
E
DE = 9.82 cm
C
DC = 1.56 cm
Regular Octagon (Rotate points 45 degrees.)
Perimeter P1 = 8.00 cm
Circumference
Perimeter P1 = 3.06 cm
Circumference
DE = 3.14 cm
DC = 0.50 cm
G
P
H 1 FB
I D A
J CE
DE = 8.21 cm
P1
G
F
H
B
DC = 1.31 cm
D
I
A
J
CE
Regular Twenty-gon (Rotate points 18 degrees. I changed this to twenty instead of
seventeen to avoid rounding errors.)
Perimeter P1 = 7.53 cm
Circumference
Perimeter P1 = 3.12 cm
Circumference
DC = 0.50 cm
DE = 7.56 cm
D
DC = 1.20 cm
DE = 3.14 cm
D P1
E
C
P1
E
C
As the number of sides approach infinity, the perimeter approaches 3.14 when the
diameter is one in the circumscribed circle, which is the circumference of the circle.
As the number of sides approach infinity, the perimeter approaches the circumference of
the circle. The difference between the two decreases and the ratio of the perimeter to the
circumference approaches 1.
Extensions of the Problem
Is a circle a polygon? If it is, explain why. If it is not, explain what type of polygon it
most closely resembles.
A circle is not a polygon. Note the following definitions:
Circle: The set of all points in a plane that are the same distance, called the radius, from
a given point, called the center.
Polygon: A closed figure formed by three or more line segments.
A polygon requires line segments. A circle does not have line segments. From an
algebra perspective the only slope possible is from a line tangent to the circle or the
chords through two points of the circle. Each side of a polygon has a distinct linear
slope. The distance from the center of a polygon is not the same to all other points on the
polygon, but it is in a circle. A polygon with infinity sides (increasing in number) would
most likely resemble a circle.
Author & Contact
Nicole McDowell
nmcdowell@rockdale.k12.ga.us