Ratio of Circumference to Diameter Activity
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Early peoples probably drew circles by rotating a stick or a taut length of rope or string about a center point. They may even have formed circles by pulling a loop of rope or string out as far as it would go in all directions, then tracing around it. (See the figure below.)
Once
a circle was formed, they may have measured the circumference and the diameter
of the circle, and noticed that, no matter how large or small the circle, the
ratio of the circumference to the diameter always seemed to be the same, a
little larger than 3. That is, if C is the circumference and D is
the diameter, then the quantity C/D is a constant number a little larger
than 3. Today, we call this constant quantity p, so that 
, or C
= pD. Since D = 2r, where r is the radius of
the circle, this last formula yields our usual expression for the circumference
C of a circle of radius r, 
.
Practitioners of mathematics in ancient Egypt, Mesopotamia, China, and India knew that the ratio C/D is constant and a bit larger than 3. They often used 3 as an estimate for C/D or p, so that the relationship between the circumference and diameter of a circle, for them, was C = 3D. They almost certainly knew that 3 was less than the actual value of p, but probably used it out of convenience. Researchers have yet to determine whether or not the Mayas, Aztecs, Incas, and other Native American peoples knew that C/D is constant, but, given the sophisticated astronomical knowledge and architecture of many of these groups, it seems likely that they did know that C/D is always the same.
In Problem 1, you’ll check for yourself that C/D always is the same. In Problems 2-5, you’ll compute some values for C/D used in very early times.
1. Measure with a measuring tape
(or with a string and a ruler) the circumference and diameter of various
circular objects (e.g. jar lids, pan lids, plates, clocks). Record the
circumference and diameter of each object in the table below, then compute C/D
for each entry. Use your calculator's value for p to compute the
error in each of your approximations---that is, to compute 
.
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C |
D |
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2. Suppose you find the following instructions in an ancient document. "Build a circular enclosure measuring 6.25 meters around and 2 meters across." What value is being used for p ? Explain.
3. The following description appears in the Chinese mathematical manual, Nine Chapters on the Mathematical Art, originally composed about 200 BCE (Katz, 20). "One has a round field; the circumference is 30 steps, the diameter 10 steps." What value is being used for p ? Explain.
4. The following description appears in the Hebrew Bible (Old Testament of the Christian Bible) in I Kings 7:23. The passage concerns events that occurred during the reign of Solomon in about 950 BCE (Katz, 20). "And he made a molten sea of ten cubits from brim to brim, round in compass . . . and a line of thirty cubits did compass it round about." What value is being used for p ? Explain.
5. The following description appears in the Aryabhatiya, written by the Indian mathematician Aryabhata, who was born in 476 CE (Berggren, et al, Appendix 1, 679). "Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000." What value is being used for p? Explain.
Instructor Notes
Objective: Students will discover that the ratio C/D is a constant, namely p.
Materials: For each group of students, provide a measuring tape (or a string and a ruler) and various circular objects (e.g. jar lids, pan lids, pizza pans, plates, clocks) for measuring the circumference and diameter.
How to Use: Students should work in groups of 2-4 students each. As a modification, you could have students discover that C/D is a constant, approximately equal to 3.14, and have them complete Problem 1 without mention of p as they work. If you choose to take this approach, be sure to delete the last sentence of Problem 1 and the last column of the table given there from your initial instructions to the students. Have them read the Background Information and read and complete the remaining problems after completing Problem 1.
Students also could take measurements for Problem 1 using as units of measurement their own personal finger, palm, or even cubit (elbow to fingertips). Or, draw large circles on the classroom floor or in the parking lot, and have students measure them using as their units of measurement their own personal foot (shoe) lengths.
Related Activity: Hold a Pi Day celebration on March 14, preferably at 1:59 p.m.
Solutions:
1. You might suggest that students average their estimates of p to see if they can obtain a better approximation of p.
2. p = C/D = 6.25/2 = 3.125
3. p = C/D = 30/10 = 3
4. p = C/D = 30/10 = 3
5. p = C/D = 62832/20000 = 3.1416
References: Activity from Lengths, Areas, and Volumes, by J. Beery, C. Dolezal, A. Sauk, and L. Shuey, in Historical Modules for the Teaching and Learning of Secondary Mathematics, Mathematical Association of America, Washington, D.C., 2003.
Berggren, L., J. Borwein, and P. Borwein (eds.), Pi: A Source Book, Springer-Verlag, New York, 1997.
Katz, Victor J., A History of Mathematics: An Introduction, Addison-Wesley, Reading, Massachusetts, 1998.