Investigating the Relationship Between
Circumference and Diameter
Purpose: to determine the relationship between the
circumference and the diameter of cylindrical objects; to
determine the accuracy of measurement; to compare accuracy
and precision
Procedure:
Materials:five cylindrical objects, ie. metal tubing, salt
containers, etc
string (non-elastic)
metric ruler
graph paper or graphing program
1.
2.
3.
4.
Obtain at least five cylindrical objects for
measurement.
Construct a data table that lists the name of the
object, the circumference (cm) and the diameter
(cm). C and d MUST be measured…not calculated.
Make measurements using an estimated digit with the
school metric ruler (Each measurement should have
two decimal places in cm.)
Carefully measure the diameter and the circumference
of each object and record.
Questions/Analysis:
1.
2.
3.
For each object, divide the circumference by the
diameter. Show your work and your answers. Do you
notice any pattern or similarities between the
resulting numbers? What unit does the resulting
number have?
Construct a graph by hand or by using a graphing
program such as Graphical Analysis. (Excel can also
be used, but gives no statistics on the resulting
line formed.)
To construct by hand, place the circumference on the
y-axis and the diameter on the x-axis.
To
determine the scale for the axis, follow this
procedure: For the y-axis, first count the number
of boxes on the y-axis; then divide the highest
circumference value in the data table by the number
of boxes. Round UP to the nearest workable value.
For example, if when divided the answer is
0.033…round to 0.05. If the value is 1.7…round to
4.
5.
6.
7.
8.
9.
10.
2. Usually multiples of 1, 2 and 5 work best for
counting. This is the number that EACH BOX will
represent. But, label the lines, not the box. DO
NOT label every line since it will look crowded;
instead label ~every 5th line.
To construct the x-axis (diameter), count the number
of boxes on the x-axis and then divide the highest
diameter by the number of boxes. Finish as in Step
3.
Using the method above will allow use of nearly all
of the axis. Graphs do not look good if they are
scrunched in one corner or if the data points go off
the page.
Plot the five points from the data table on the
graph paper.
Make a best-fit line by taking a ruler and drawing
the BEST AVERAGE line through the points. You DO
NOT have to start at zero, although for this graph
the line should end up going through (0,0). End the
line near the last data point. THE LINE DOES NOT
HAVE TO TOUCH ANY POINT YOU HAVE COLLECTED! This is
called a REGRESSION line, which is an average of all
data points. Some points should lie above and some
below the line.
Give the graph a title; usually the format Y vs. X,
in this case Circumference vs. Diameter. Line
graphs in science are generally titled this way.
Make sure that each axis is properly labeled and
contains units: Circumference (cm) and Diameter (cm)
Make an assessment of your graph using the following
rubric:
Criteria
Variables
0 points
The
independent
and dependent
variables are
not shown.
Label
The axes are
not labeled.
1 point
The
independent
and dependent
variables are
present, but
they are
improperly
placed on the
graph.
The variable
name is
shown, but
the unit is
missing.
2 points
The
independent
and dependent
variables are
placed on the
correct axes.
The correct
variable name
and unit are
shown on each
axis.
Title
The graph
does not have
a title.
The graph has
a title, but
the title is
not correctly
written.
Scale
No scale is
shown.
Plots
Data points
do not
accurately
present the
information.
Graph Line
The graph
line is not
shown on the
graph.
The scale is
shown, but it
is
inappropriate
for the data.
Data points
are shown,
but one or
more points
is/are
improperly
placed.
The graph
line is
shown, but
the position
of the line
does not
accurately
represent the
data.
11.
12.
The title of
the graph
follows the
generic form
of "y versus
x".
The scale for
each axis is
appropriate
for the data.
All data
points are
clearly and
accurately
plotted.
The graph
line
correctly
represents
the data.
Find the slope of the line on the graph by using the
formula y/x or Rise/Run. What are the units for
this slope?
Calculate a percentage error using the following
formula:
|EXP – THEO|
THEO
x 100
In this case, the EXPerimental value is the slope
you found.
The THEOretical value is 3.14.
Absolute value is used so that all percentage errors
are positive.
13.
14.
Why is the theoretical value for this lab 3.14?
What error is associated with measuring diameter?
DO NOT INCLUDE HUMAN ERROR LIKE MISREADING THE
RULER….WE ALWAYS ASSUME SOME HUMAN ERROR!
15.
16.
17.
18.
What error is associated with measuring
circumference? Only include error associated with
instruments/tools used.
The percentage error you found is called the
ACCURACY…how close the experimental value is to the
standard or theoretical value. PRECISION is the
reproducibility of the measurements. When you
divided C by d in question 1, were the values close
to each other? If they were close, precision was
good; if they were random, precision was poor.
Describe your precision.
Using a dartboard as an example, describe how a dart
thrower, using three throws, could have the
following:
(A) good accuracy and good precision
(B) poor accuracy and good precision
(C) poor precision and poor accuracy
In this lab, your accuracy could be good with the
precision being poor. Explain using your graph to
support your argument.
Conclusion:
Write a paragraph that summarizes the results and addresses
the purpose of the lab. There are three parts of the
purpose to address in this conclusion. DO NOT RESTATE THE
PURPOSE AND DO NOT LIST ALL THE DATA!