CUPOL: Determining p

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It is well known that the circumference, C, of any circle with diameter, d, may be determined by the formula,

C = p d

Eq. 1

The value for p may be determined experimentally by plotting the circumference of several circles versus their diameter. Recall that the general equation of any straight line is given by

y = mx + b

Eq. 2

where m is the slope and b is the y-intercept of the line. If the line passes through the origin, then b = 0, and Equation 2 becomes

y = mx

Eq. 3

Equation 1 has the identical form of Equation 3. Therefore plotting several circles' circumferences versus their diameters will yield a slope that is equivalent to p. Notice that the y-intercept should be identically zero. When calculating the percent error, assume the value for p to be 3.14159.

• Coins (one each: penny, nickel, dime, quarter and half dollar)
• Transparent sticky tape
• Meter stick
• Vernier caliper
• Graph paper
• Straight edge

Figure 1.

[Click on image to enlarge it.]

View the following video clips to measure and record each coin's circumference.

Measure the penny's circumference. [1:14, 13.8 Mb]
Measure the nickel's circumference. [0:41, 5.6 Mb]
Measure the dime's circumference. [0:41, 7.6 Mb]
Measure the quarter's circumference. [0:39, 7.3 Mb]
Measure the half dollar's circumference. [0:35, 6.2 Mb]

Figure 2.

[Click on image to enlarge it.]

View the following video clips to measure and record each coin's diameter:

Measure the penny's diameter. [0:28, 5.4 Mb]
Measure the nickel's diameter. [0:31, 5.8 Mb]
Measure the dime's diameter. [0:30, 5.7 Mb]
Measure the quarter's diameter. [0:30, 5.8 Mb]
Measure the half dollar's diameter. [0:31, 6.0 Mb]

• If you are creating the graph by hand you may wish to view our graphing tutorial.
• If you are using Excel to store the data and make the graph, you may want to view our Excel tutorial.

From your graph, determine the equation of the line based on Equation 1.

1. Compare your experimental value for p with the accepted value of 3.14159 by calculating the percent error between the two values.
2. The careful student should be able to achieve a percent error as low as 0.6%!

Coin	Circumference (cm)	Diameter (cm)
Penny
Nickel
Dime
Quarter
Half Dollar

1. List the sources of error in this experiment.
2. If we had used string, rather than sticky tape, to determine the coins' circumferences, how would the results differ?
3. What does the graph's y-intercept represent? Was the y-intercept close to zero? Is this what you expected?
4. If you were to measure the diameter of a button to be 2.28 cm, describe how you would use your graph to determine the button's circumference. Determine the button's circumference using your graph. What is the value of the circumference using the formula C = p d?
5. The density, r, of a solid is found by dividing the object's mass by its volume, or, r = m/V. If the masses and volumes of several objects are given and you are told that each object is made from the same material, describe how to construct a graph to determine the density of this material. What is plotted along each axis? Sketch what the graph should look like. What are the units of the slope? (Hint: Algebraically rearrange the density formula to make it take the form y = mx + b.)
6. The area of a circle with radius, r, is given by the formula A = p r². Given the area and radius of several circles, describe how you would construct a graph to determine the value for p. What is plotted along each axis? Sketch what the graph should look like. What are the units of the slope? What should the y-intercept be? (Hint: This graph is not a parabola.)