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 want to find out the relationship between circumference and diameter so that I can tell
how the circumference of a circle changes when doubling, tripling, and halving the
diameter. I want to find out why a change occurs when changing the diameter.
Plans to Solve/Investigate the Problem
Prediction: the circumference changes at the same rate as the diameter change.
To begin the problem, I will construct a circle. I will then determine the circumference
and diameter. Constructing these figures in GSP allows me to measure the circumference
with the measure, circumference tool. I can use a formula to determine the diameter;
d = 2r. I will double the diameter to see what happens to my circumference, triple the
diameter, and halve the diameter. I will examine my measurements to determine why a
change occurs when changing the diameter. Lastly, I will prove whether my prediction is
right or wrong, and tell why.
Investigation/Exploration of the Problem
1.
Using the compass tool in GSP, construct a circle of any size.
B
A
2. Select the circle. On the menu bar, select Measure  Circumference. Remember
that Circumference =  (diameter)  C =  d
3. Select vertex A and B. On the menu bar, select Construct  Segment
4. Measure the length of AB . This measurement is the radius of the circle.
5. To find the diameter, the equation for diameter must be known. d = 2r
Select AB . On the menu bar, Transform  rotate  fixed angle at 180°.
6. Construct vertex C.
B
C
A
7. Construct line segment BC by selecting vertex’s B and C. On the menu bar,
Construct  segment.
8. Measure the length of BC . This is the diameter of the circle.
My Measurements:
Radius
AB = 1.72 cm
Circumference
AB = 10.81 cm
Notice from my measurements that the
radius, AB is half of the diameter, BC .
m AB = 1.72 cm
m AB2 = 3.44 cm
m BC = 3.44 cm
m AB2 = 3.44 cm
9. To double the diameter, select BC . (This is now your radius because the diameter
you are aiming for is twice the diameter of circle A (3.44), which would be 6.88,
which means that half of 6.88, 3.44 becomes the radius of the circle you are about
to construct.) Using the point tool, place a random point anywhere outside circle
A. Construct  Center by circle + radius.
B
A
C
10. Name the circle that you just constructed D using the text tool. Place a random
point (vertex), E, anywhere on circle D. Select Vertex D and E and construct a
segment. You have now constructed the radius of circle D.
11. To get the diameter, rotate DE around center (vertex) D 180°.
My Measurements:
Radius c1 = 3.44 cm
Notice from my measurements
that the radius, 3.44cm, is half of
the diameter, 6.88cm.
Circumference c1 = 21.62 cm
m DE = 3.44 cm
m FE = 6.88 cm
m DE2 = 6.88 cm
A relationship is now seen between the circumference and diameter…
So far, my prediction proves to be true; the circumference changes at the same rate as the
diameter change.
Circumference
AB2 = 21.62 cm
Note: As the diameter doubled,
the circumference doubled.
12. To triple the diameter of the circle, select and copy EF . I created a midpoint, H,
and rotated DF 180°. The diameter has now tripled the original diameter.
H
E
D
F
G
My Measurements:
Circumference HE = 32.43 cm
Radius HE = 5.16 cm
Notice: HE and HG are the same length and
have the same radius from center H, proving H the
midpoint of the circle. Multiplying the original
diameter by three proves that circle H has tripled
in diameter.
m HE = 5.16 cm
m HG = 5.16 cm
m EG = 10.32 cm
m HE2 = 10.32 cm
m BC3 = 10.32 cm
A relationship is seen again between the circumference and the diameter…
Circumference
AB3 = 32.43 cm
As the diameter tripled, the circumference tripled.
The original circumference of circle A, 10.81 cm
has been multiplied by three, and is now 32.43 cm.
The diameter of beginning circle A, 3.44 cm, tripled
is the diameter of circle HE, 10.32 cm.
m BC3 = 10.32 cm
13. When halving the diameter, select and copy AB . Construct midpoint L.
Construct circle by center + radius around center L and vertex J.
L
J
K
My Measurements:
Circumference LJ = 5.41 cm
Radius LJ = 0.86 cm
m LJ = 0.86 cm
m LK = 0.86 cm
m KJ = 1.72 cm
m LJ2 = 1.72 cm
m BC
2
= 1.72 cm
The diameter of the circle is halve the diameter of
circle A, as seen when the diameter of circle A,
BC is divided in half and equals the diameter of
LJ .
This case also corresponds with my prediction.
 Circumference
2
Circumference
m BC
2
= 1.72 cm
AB
= 5.41 cm
LJ = 5.41 cm
m KJ = 1.72 cm
The circumference of beginning A equals
the circumference of LJ when divided in
half. The diameter of A equals the diameter
of LJ when divided in half  1.72 cm.
LJ
Looking at the constructions in this write-up, my prediction holds true; the circumference
changes at the same rate as the diameter change. There is a direct variation between the
circumference and the diameter. The relationship between circumference and diameter is
constant.
Extensions of the Problem
Camera lenses have a diaphragm that controls the aperture, the opening which
allows light to pass through to the film.
For example, in the picture above, the outer circle is the lens and the inner circles
are the diaphragms. Note that the size (area) of the aperture can change. A larger
aperture allows more light to pass through. The size of the aperture is determined
by the f numbers (also called f stops). A typical 50 mm lens has the following f
stops: f 2.8, f 4, f 5.6, f 8, f 11, f 16. The diameter of the aperture is determined by
the fraction:
50 mm
f stop
Compare the amounts of light passing through a 50 mm lens for various pairs of f
stops. How much more light passes through f 4 than f 16? Through f 2.8 than f 5.6?
Try to generalize any patterns you find. Test your conjectures on a 120 mm lens with
the same f stops.