How to Write a Good Lab Report
Sample Lab Instruction
Experimental Investigation of C/D
Introduction: How is the circumference of a circle related to its diameter? In this lab, you design
an experiment to test a hypothesis about the geometry of circles. This activity is an introduction
to physics laboratory investigations. It is designed to give practice taking measurements,
analyzing data, and drawing inferences without requiring any special knowledge about physics.
Equipment (per group):
• Metric ruler
• Vernier calipers
• At least 5 objects with diameters ~1 cm to ~10 cm: (penny, marble, “D” cell, PVC
Design an experimental procedure to test the following hypothesis:
Hypothesis: The circumference (C) of a circle is directly proportional to its diameter (D).
Make sure you record what you do as you do it, so that the procedure section of your report
accurately and completely reflects what you did. Some helpful hints for taking and recording
data are in the lab tips and in the grading rubric.
Note: As the semester progresses, you will be expected to take more and more responsibility for
deciding how to analyze your data. Drawing valid inferences from data is a vital skill for
engineers and scientists. The instructions for analyzing data for most labs will not be as detailed
as the instructions below.
• Numerical Analysis: Calculate the ratio C/D for each object. Estimate the precision of
each value of C/D.
• Graphical Analysis: Use Excel to construct a graph of C versus D. Use Excel to display
the equation of the best fit line through your data. Use the LINEST function to estimate
the uncertainty in the slope and intercept of the best fit line. Make sure you interpret the
meaning of both the slope and intercept. A checklist for graphs is in the grading rubric.
• Questions to consider:
How do your calculations and graph support or refute the hypothesis?
Does your graphical analysis agree with your calculations?
Do your results for the C/D ratio agree with accepted theory?
A sample lab report for this activity is provided as an example for you to follow when writing
future lab reports.
Sample Lab Report: Experimental Investigation of C/D
In this investigation, we examined the hypothesis that the circumference (C) and diameter (D) of
a circle are directly proportional. We measured the circumference and diameter of five circular
objects ranging from 2 cm to 7 cm in diameter. Vernier calipers were used to measure the
diameter of each object, and a piece of paper was wrapped around each cylinder to determine its
circumference. Numerical analysis of these circular objects yielded the unitless C/D ratio of 3.14
± 0.03, which is essentially constant and equal to pi. Graphical analysis lead to a less precise but
equivalent estimate of 3.15 ± 0.11 for this same ratio. These results support commonly accepted
geometrical theory which states that C = pD for all circles. However, only a narrow range of
circle sizes were analyzed, so additional data should be taken to investigate whether the constant
ratio hypothesis applies to very large and very small circles.
Five objects were chosen such that measurements of their circumference and diameter could be
obtained easily and would be reproducible. Therefore, we did not use irregularly shaped objects
or ones that could be deformed when measured. The diameter of each of the 5 objects was
measured with either the ruler or caliper. The circumference and diameter of each object was
measured with the same measuring device in case the two instruments were not calibrated the
same. The circumference measurement was obtained by tightly wrapping a small piece of paper
around the object, marking the circumference on the paper with a pencil, and measuring this
distance with the ruler or caliper. The uncertainty specified with each measurement is based on
the precision of the measuring device and the experimenter’s estimated ability to make a reliable
• “D” cell battery, 2 short pieces of PVC pipe, tomato soup can, penny coin
• Metric ruler with millimeter resolution
• Vernier caliper with 0.05 mm resolution
Object Description Diameter(cm) Circumfer(cm) Measuring Device
Penny coin 1.90 ± 0.01 5.93 ± 0.03 Vernier caliper, paper
“D” cell battery 3.30 ± 0.02 10.45 ± 0.05 Vernier caliper, paper
PVC cylinder A 4.23 ± 0.02 13.30 ± 0.03 Vernier caliper, paper
PVC cylinder B 6.04 ± 0.02 18.45 ± 0.05 Plastic ruler, paper
Tomato soup can 6.6 ± 0.1 21.2 ± 0.1 Plastic ruler, paper
The C/D value for the penny is (5.93 cm)/(1.90 cm) = 3.12 (no units). The precision of the ratio
can be estimated using the error propogation formula:
Results for all five objects are given in the table below.
Object Description Diameter(cm) Circumference(cm) C/D calculated (no units)
Penny coin 1.90 ± 0.01 5.93 ± 0.03 3.12 ± 0.02
“D” cell battery 3.30 ± 0.02 10.45 ± 0.05 3.17 ± 0.02
PVC cylinder A 4.23 ± 0.02 13.30 ± 0.03 3.14 ± 0.02
PVC cylinder B 6.04 ± 0.02 18.45 ± 0.05 3.06 ± 0.01
Tomato soup can 6.6 ± 0.1 21.2 ± 0.1 3.21 ± 0.05
Average C/D = 3.14 ± 0.03, where 0.03 is the standard error of the 5 values.
From this empirical investigation, the average C/D ratio is 3.14 ± 0.03 (no units). This ratio
agrees with the accepted value of pi (3.1415926…). The uncertainty associated with the average
C/D ratio is the standard error of the five C/D values, which is equal to the standard deviation
(0.06) divided by the square root of N, which in this case is 5 since there were five
While the five C/D values do not agree within their estimated uncertainties, the variation
between these values is relatively small (only about 0.06/3.14 = 2%), which suggests that the
C/D ratio is a constant value. The reason for the imperfect agreement may be that the individual
uncertainties were underestimated or perhaps is a consequence of the “paper” method used for
measuring the diameters of the object. The paper may have slipped while we made the mark, but
this “slip effect” should only be a random error, which would not affect the average value of our
measurements for C, since there is no reason to believe that the paper would have consistently
slipped in the same direction (either too high or too low) every time.
Another way to visualize and calculate this constant circle ratio is by graphing the circumference
versus diameter for each object. Graphs are especially useful for examining possible trends over
the range of measurements.
If C is proportional to D, we should get a straight line through the origin. From our numerical
results, we would expect the slope of the C vs. D graph to be equal to pi. The slope of the best fit
line is (3.15 ± 0.11), which is equal to pi within its uncertainty. The intercept is essentially zero:
(-0.05 ± 0.5). The R squared statistic shows that the data all fall very close to the best fit line. If
all the data lie exactly on the fitted line, R squared is equal to 1. If the data are randomly
scattered, R squared is zero. With an R^2 value of 0.997, our linear equation appears to fit the
data very well.
Our results support the original hypothesis for 5 circles ranging in size from 2 cm to 7 cm in
diameter. The C/D ratio for our objects is essentially constant (3.14 ± 0.03) and equal to pi. The
specified uncertainty is the standard error of the C/D ratio for the five objects. Graphical analysis
also supports the “directly proportional” hypothesis. The line has an intercept (-0.05 ± 0.5) that is
equal to zero within the uncertainty and a slope (3.15 ± 0.11) equal to pi . The larger uncertainty
from the graphical analysis suggests that the random measurement errors may be larger than
estimated in the numerical analysis. A more extensive investigation of this C/D relationship over
a wider range of circle sizes should be performed to verify that this ratio is indeed constant for all
The uncertainty in the measurements could be due to the paper-wrapping method of measuring
the circumference, circles that may not be perfect, and the limited precision of the measuring
devices. The use of paper to measure the circumference was probably the most significant source
of uncertainty. It is unlikely, however, that this measurement technique biased our results, since
the technique probably gave measurements of C that were too high in some cases and too low in
The C/D ratio for a perfect circle was defined long ago by the Greek symbol: pi = 3.14159… Our
measured value appears to be consistent with the accepted value of pi within the limits of our
experimental uncertainty. This unique C/D ratio has many important applications wherever
circles or spheres are encountered.
Lab Report Format
Lab reports must be typed or printed clearly in ink. Each report must include a cover page, an
abstract, a summary of your experimental results (including your original data sheet and any
additional notes, tables, or graphs), and a discussion of your findings. See the descriptions below
for what should be included in each section.
• Title of experiment
• Lab section
• Date the experiment was performed
• Your name
• Your partner’s name (identified as such)
The abstract is a concise summary of the lab report. A good abstract should state the purpose,
procedure, principal results, conclusion, and implications of the lab in a single paragraph that is
generally 100 to 200 words in length (use your word processor’s word count tool to check
A complete scientific lab report has an introduction that gives the context for the experiment, the
background theory, and a description of the experimental procedure and equipment used. For
simplicity and brevity, you are not required to include this section, but you may do so if you
prefer. In cases where a particular lab does not have a prescribed procedure, or you used a
procedure that was significantly different than the one described in the lab manual, you should
clearly explain what you actually did either in the introduction or discussion sections.
DATA AND RESULTS
The results of your experiment must be well organized and easy to read. When appropriate,
tables or graphs should be used to present data and results. Graphs must be properly constructed
(with a computer or by hand, as directed) with descriptive titles, labeled axes with relevant units,
and calculated parameters properly interpreted (e.g. What do the slope and intercept represent?)
All measured values must have four critical parts:
1. A label (word or symbol) that clearly identifies the measured value
2. The numerical value for the measurement (rounded to be consistent with the uncertainty)
3. A reasonable estimate of the uncertainty associated with the measurement
4. An appropriate unit of measure (SI units are usually preferred)
Sample calculations, including an analysis of the experimental uncertainties, should be shown for
any derived or calculated values as appropriate. Your original, unaltered data sheet must be
included either in the data section or as an appendix.
In the discussion section, summarize the results you obtained, and then discuss any discrepancies
between your results and what was expected according to the given theoretical predictions or
your own hypotheses. Did the experimental results agree with your predictions or the findings
from other lab groups? If not, what is the most likely reason for the discrepancy? Remember to
consider the uncertainty of your results when determining agreement. Identify the primary source
of error in your results and justify your answer based on your uncertainty estimates. (Note:
General statements without justification and explanation are not acceptable. “Human error” is
not an acceptable source of error!) How could you improve the quality of your measurements
with the available equipment? What did you learn or discover from this lab? The discussion
section for most labs should be about one to two pages in length. Remember that your discussion
will be graded on the quality of your explanations, not the quantity.