InterMath
Title
Changes in the Circumference
Problem Statement
What happens to the circumference of a circle if you double the diameter? If you triple
the diameter? If you halve the diameter? As the diameter increases (or decreases) in
measure, how does the circumference change? Why does this change occur?
Problem setup
I am trying to determine what happens to the circumference of a circle as I increase and
decrease the diameter. I know that to find the diameter of a circle I must multiply the
radius by two. The radius is the distance from the center of the circle to one point on
the outside of the circle. The diameter is the distance from one point on the outside of
which passes through the center of the circle to reach another point on the on the other
side of the circle. I know the circumference will change because the formula for finding
ci rcumfe rence=2r
ci rcumfe rence=d
the circumference of a circle is
These two formulas are different because one uses the radius and one uses the
diameter. The diameter and radius are closely related because, the diameter is found
by multiply the radius times 2.
Plans to Solve/Investigate the Problem
Prediction: as diameter increases, circumference will increase, double diameter &
circumference will double, triple diameter and circumference will triple, and half
diameter and circumference will half
First I plan to construct the initial circle in GSP using the compass tool with a radius of 1
cm. I will then multiply that by two to get my diameter of 2 cm. Then I will measure the
circumference of the circle. I will continue this process until I have doubled, tripled, and
halved the diameter of my original’s circle’s diameter. I will then compare the results
and compare the changes in circumferences with the changes in diameters. Below are
some formulas and facts that might be helpful when doing this investigation.
Ci rcu mfere nce =2r
or
Ci rcu mfere nce = d
Di amete r=2 x radi us
.
Investigation/Exploration of the Problem
In order to discuss how I went about solving this problem I will do it in a step by step
process.
1. Create a circle: use the compass tool in GSP; construct a radius from the
center point of the circle to a point on the outside of the circle
J
K
Radius
KJ = 1.01 cm
2. We know that the diameter of a circle connects to points on a circle by
passing through the center of the circle. Therefore, to get the diameter, we
simply multiply the radius times 2. The resulting diameter should be 2.02 cm.
Ci rcumference
KJ = 6.32 cm
Radi us KJ = 1.01 cm
JL = 2.01 cm
J
This circle represents my initial circle. From
this circle and diameter, I will be doubling,
tripling, and halving the diameter to create my
other 3 circles.
K
L
JL represents the diameter. Since GSP rounds, the measurements are never
exact. This is why I ended up with a measurement of 2.01 cm. However, it the
measurements were exact I would have had a diameter of 2.02 cm
3. Next, I will double the diameter. Below is the illustration with the circle’s
diameter doubled. I will give the measurements for the radius, diameter, and
circumference.
Ci rcumfere nce
Rad ius
AB = 12.63 cm
AB = 2.01 cm
BC = 4.02 cm
B
A
C
BC represents the diameter with the measurement of 4.02 cm. As I increased
the diameter the circumference increased. The diameter doubled from the
previous circle and the circumference also doubled from the original circle.
4. Below I will show a comparison between the original circle and the circle
whose diameter has been doubled.
Ci rcumfere nce
Rad ius
Ci rcumference
KJ = 6.32 cm
Radi us KJ = 1.01 cm
JL = 2.01 cm
J
AB = 12.63 cm
AB = 2.01 cm
BC = 4.02 cm
B
A
K
L
C
The left illustration shows the original circle and the right illustration shows the
circle whose diameter has been doubled. As you can see, the circumference has
almost exactly doubled as well as the diameter. The original circle’s
circumference (circle KJ) was 6.32 cm and the circle whose diameter was
doubled (circle AB) circumference was 12.63 cm. Again, I think it is important to
not that GSP rounds, therefore the numbers are not exact and hence that is why
my numbers are a little off by one hundredths in most places.
5. Next, I will triple the diameter of the original circle. Below is the illustration
with the circle’s diameter tripled. I will give the measurements for the radius,
diameter, and circumference.
Ci rcumferen ce
Rad ius
ED = 18.96 cm
ED = 3.02 cm
DF = 6.04 cm
D
E
F
DF represents the diameter with the measurement of 6.04 cm. As I increased
the diameter the circumference increased. The diameter tripled from the original
circle and the circumference also tripled from the original circle.
6. Below I will show a comparison between the original circle and the circle
whose diameter has been tripled.
Ci rcumferen ce
Rad ius
ED = 18.96 cm
ED = 3.02 cm
DF = 6.04 cm
D
Ci rcumference
KJ = 6.32 cm
Radi us KJ = 1.01 cm
JL = 2.01 cm
E
J
K
L
F
The left illustration shows the original circle and the right illustration shows the
circle whose diameter has been tripled. As you can see, the circumference has
almost exactly tripled as well as the diameter. The original circle’s circumference
(circle KJ) was 6.32 cm and the circle’s whose diameter (circle ED) was tripled
circumference was 18.98 cm. Again, I think it is important to not that GSP
rounds, therefore the numbers are not exact and hence that is why my numbers
are a little off by one hundredths in most places.
7. Next, I will half the diameter of the original circle. Below is the illustration with
the circle’s diameter halved. I will give the measurements for the radius,
diameter, and circumference.
Ci rcumfere nce
Rad ius
TU = 3.16 cm
TU = 0.50 cm
UV = 1.01 cm
TU
V
UV represents the diameter with the measurement of 1.01 cm. As I decreased
the diameter the circumference decreased. The diameter halved from the
original circle and the circumference also halved from the original circle.
8. Below I will show a comparison between the original circle and the circle
whose diameter has been halved.
Ci rcumference
KJ = 6.32 cm
Radi us KJ = 1.01 cm
JL = 2.01 cm
Ci rcumfere nce
J
Rad ius
TU = 3.16 cm
TU = 0.50 cm
UV = 1.01 cm
K
L
TU
V
The left illustration shows the original circle and the right illustration shows the circle
whose diameter has been halved. As you can see, the circumference has almost
exactly halved as well as the diameter. The original circle’s circumference (circle KJ)
was 6.32 cm and the circle whose diameter was halved circumference (circle TU) was
3.16 cm. Again, I think it is important to not that GSP rounds, therefore the numbers are
not exact and hence that is why my numbers are a little off by one hundredths in most
places.
9. The illustrations below show each of the circles. The first being the original (circle
KJ), the second is the circle whose diameter was doubled (circle AB), the third is the
circle whose diameter was tripled (circle ED), and the fourth is the circle whose
diameter was halved (circleTU). As I mentioned above, as the diameter increased, the
circumference increase and as the diameter decreased the circumference decreased.
The reason for this is that the formula for finding the circumference of a circle is
Ci rcu mference =2r
or
Ci rcu mference = d
. By looking at these formulas we know that we are multiplying. First, I
think I must make it clear the relationship between these formulas. Diameter is found
by multiplying the radius times 2. Therefore this is why we do away with the 2 and r in
the first formula and end up with the second formula. Each formula would work
because they are saying the same exact thing.
Therefore, we know when we multiply 2 numbers we know the higher the
numbers the higher the product will be. We also know the lower the numbers, the lower
the product will be. Therefore as the diameter increases the circumference increases
and as the diameter decreases the circumference decreases.
Ci rcumferen ce
Rad ius
ED = 18.96 cm
ED = 3.02 cm
DF = 6.04 cm
D
Ci rcumfere nce
Rad ius
Ci rcumference
KJ = 6.32 cm
Radi us KJ = 1.01 cm
JL = 2.01 cm
AB = 12.63 cm
AB = 2.01 cm
BC = 4.02 cm
B
E
J
A
K
L
Ci rcumfere nce
Rad ius
C
F
TU = 3.16 cm
TU = 0.50 cm
UV = 1.01 cm
TU
V
Extensions of the Problem
What happens to the area of a circle if you double the diameter? If you triple the
diameter? If you halve the diameter? As the diameter increases (or decreases) in
measure, how does the area change? Why does this change occur?
1. Using the same illustrations and numbers, I wonder what will happen to the
area of the circle if I double, triple, and half the diameter of the circle.
2. First I must find out what the formula for finding the area of a circle is. Below
is the formula for finding the area of a circle. The formula states pi times
radius squared equals the area of a circle.
Area of a ci rcl e =  x radius2
3. We know that the diameter of a circle is the radius multiplied times 2.
Therefore to get the radius of the circle we need to divide the diameter by 2 or
in this case, GSP gives us the radius measurement.
4. The illustrations below show each of the four circles. The measurements for
the radius, diameter and area are given for each circle.
Rad ius
ED = 3.02 cm
DF = 6.04 cm
Area
ED = 28.62 cm 2
D
Rad ius
AB = 2.01 cm
BC = 4.02 cm
Area
AB = 12.70 cm 2
B
Rad ius KJ = 1.01 cm
JL = 2.01 cm
E
KJ = 3.18 cm 2
Area
J
A
K
L
.
Radi us
C
F
TU = 0.50 cm
UV = 1.01 cm
Area TU = 0.79 cm 2
TU
V
5. Here, we see the same thing happening. As the radius increases, the area
increases and as the radius decreases, the area decreases. If we notice, in
the formula, the radius is squared. Therefore, the larger the number that we
square the higher our product will be. So, it is easy to see why the circle with
a radius of 1 would have a smaller area than the circle with a radius of 4.
Author & Contact
Carla McNeely, Middle Grades Education student, concentrating in English/Language
Arts and Math. I am currently a junior at Georgia College and State University.
carlalynnmc@yahoo.com
Link(s) to resources, references, lesson plans, and/or other materials
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=70
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=39