InterMath
Title
Changes in 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 start off with a circle then double the diameter. I go back to the original circle and triple the
diameter. Each time the diameter is increased or decreased, I will measure the circumference.
Plans to Solve/Investigate the Problem
I plan to construct a circle using GSP. The circle will have a set of two radii that go through the
center of the circle to create the diameter. One of the radii will be the radius. I will then find the
circumference of the circle. To show the circumference change, I will then change the diameter
to show how the circumference increases or decreases proportionally.
Investigation/Exploration of the Problem
AB = 1.00 cm
B
Ci rcumference
AB = 6.28 cm
A
C= 2 (3.14) (1.00)
21 = 6.28
2 = 6.28
AB = 1.00 cm
B
Circumference
AB = 6.28 cm
A
AC = 1.00 cm
C
AB = 2.00 cm
Ci rcumfere nce
AB = 12.54 cm
AC = 2.00 cm
B
22 = 12.57
4 = 12.57
21 2 = 12.57
A
C
2 / 2 2= 0.50
22 / 2  =2.00
Usi ng GSP, construct a ci rcl e. Lab el the center
A and the poi nt on th e ci rcl e p oint B. Make l ine
segment AB. AB is the rad ius of the ci rcl e.
Create an other point, C, and l abel i t. Since BC
goes through th e center, A, then BC is the
di ame ter. Al so, by comparin g the radi us an d
the di amete r there is eviden ce to prove the
di ame ter i s twi ce the radi us.
The circum ference (C) o f a ci rcl e i s derived by the form ula
C=2r or C=d. Ci rcu mferen ce i s th e di stance arou nd the
circle . The sym bol "r" stands for radiu s and "d" stands for
the di am eter. The radius is the distance from the center
(A) to the ou tside of the ci rcl e to any po int on the ci rcl e. In
this case, po int A to ei th er poin t B or C wil l be the rad ius.
Any poi nt on the circle to the center wi ll a lways resul t in th e
same m easure and can al ways be the radi us. A ci rcl e has
the sam e di stance from the center ou t to the edge of the
circle i f it i s a true circle and not an o va l.
The di stance from any two poi nts on the circle through the
center is call ed the diam eter. In thi s case, th e di am eter i s
the di stance from B to C, through point A. The di ameter i s
two radi i, so the di ameter wi l l alwa ys be twi ce the radi us,
since th e radiu s is the same no matte r the origi n of the
poi nt on the circle. The diam eter wil l al so al ways be the
same m easure no matte r where the p oints are on the
circle , and the radi us wil l always be half the diam eter no
m atter the l ocati on.
The Greek sym bol Pi , , i s al so in cl uded in the
circum ference formul a. Pi is the resul t of the
circum ference of any ci rcl e d ivi ded by the di ame ter of a
circle (C/d = ). The nume ri cal value of  is rounded to 3.14
al thoug h m ath emetici ans conti nue to fi nd num bers to add
to the a ctual val ue of 3.14159265 35897932384 6... The
num ber 3.14 m akes sol ving for the circumference easi er
to sim pl ify.
AB = 3.00 cm
Ci rcumferen ce
AB = 18.85 cm
AC = 3.00 cm
B
By looking at the figure to the right and the ones above, there
is obvious increase in size. The circumference is also
increasing in certain increments. From the 1 cm diameter to
the 2 cm, the circumference increases two times the original
1 cm. The diameter increases two times as well. From the 1
cm to the 3 cm, a diameter increase of three times, the
circumference also increases three times.
A
A circle with any size diameter that has an increase or
decrease of a certain amount in measure will cause the same
proportional increase or decrease in the circumference.
Below is another example of how a diameter is increased to
C
two and three times the original. When the diameter of 7 cm
decreases by half, the circumference 43.99 cm decreases to
half as well and becomes 22.01 cm. The measurements are off slightly in measuring exactly half
because of pi. Without rounding pi to 3.14 the circumferences are exactly half. Measurements
are often off by a tens or hundred place.
AB = 3.50 cm
Ci rcumfere nce
AB = 7.00 cm
AB = 22.01 cm
AC = 3.50 cm
Ci rcumfere nce
AC = 7.00 cm
AB = 10.50 cm
AB = 43.99 cm
Ci rcumferen ce
AB = 65.95 cm
AC = 10.50 cm
The increasing increments result because of the constant pi. Since there is a constant, 3.14, any
increase in the diameter, from an original circle, will result as the same amount increase in the
circumference. Pi plays a significant role in having a proportional increase. For any circle the
same principles will hold. Any increase in diameter or circumference by a certain measure or
multiple causes the same change in the other. They increase or decrease proportionally or in the
same increments.
Extensions of the Problem
Looking at a camera lens, the circumference of a circle also acts as a vital part in how much
light comes into the lens.
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.
50mm/ 4=12.5 mm
50mm/ 16=3.125mm 50 mm/ 2.8=17.8571mm 50mm/ 5.6=8.92857mm
C= 12.5 = 39.27
C= 3.125 = 9.82
C= 17.8571 = 56.10
( x 12.5) / ( x 3.125)= 4
12.5 / 3.12 5=4
C= 8.92857 = 28.05
( x 17.8571) / ( x 8.92857)= 2
17.8571 / 8.9 2857= 2
From the calculations the f stop at 4 produces a diameter of 12.5mm and a circumference of
39.27mm. However, an f stop at 16 gives a diameter of 3.125mm and a circumference of
9.82mm. Shown in the above calculations, the f stop at 4 is four times the size of an f stop at 16.
The diameter and circumference can also show how the two compare. The larger f stop always is
inversely proportional to the smaller f stop. When two f stops are compared, the diameter of the
larger f stop will be reduced in measure, therefore the circumference will also be smaller than the
lens with a smaller f stop.
A smaller aperture, shown by the f stops, allows more light to pass through because of the
diameter of the lens. Due to the circumference of the smaller interior of the lens being greater,
more light is able to pass through. The calculations show how the smaller f numbers produce a
greater circumference to show how much more light is being brought into the lens.
The same will also occur when there is a 120mm lens on the camera. When the f stops increase,
the circumference will continue to decrease and vice versa. The relationship between the
circumference and the diameter are inversely proportional because of the formula (C/d=Π).
When either of them increase or decrease the other does the opposite by the same reduction or
enlargement. Pi would not be a constant if the circumference and diameter were not inversely
proportional. Although the camera lens is using circumference, the light let in changes due to the
aperture.
Another way to look at how the radii or the diameter impacts a circle is by looking at the area.
Given a circle with a diameter of 1 cm, the area is 3.14 cm2. A circle with a diameter of 2 cm has
an area of 12.52 cm2. A circle with a diameter of 3 cm has an area of 28.37 cm2. The circles and
areas may appear to be unrelated, but the area formula for a circle describes there is a
relationship. The formula for the area of a circle is
A= r2
From the above measures, the following are seen. When the diameter is increased by two times,
the area is quadrupled. The area is four times the original because the area formula of a circle
incorporates square numbers and forms a quadratic. Since the diameter is twice the original, the
size increase of two times the original is squared. The area increases in measure by four times the
original. Therefore, a diameter with an increase three times results in an area increase of nine
times since the size change is squared.
Relationship between an original circle and twice the original: 12.52/3.14= 3.99 or 4
Relationship between the original circle and three times the original: 28.37/3.14= 9.04 or 9
The absence of pi would prevent a circle from being round and having the properties that have
been discussed. The diameter and circumference also play off of each other in order to create the
number known as pi. The constant pi allows circles to increase or decrease in diameter and
circumference at the same rate.
Author and Contact
Laura Sims
I am a junior at Georgia College and State University. I am majoring in middle grades education
with a concentration in language arts and math.
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