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Circumference vs. Diameter, page 1 of 7
Overview
Length of a Curved Line
The perimeter of a circle, the distance around its “outside,” is called the circumference. All the length measurements so far have involved the straight line distance between two points. How then would you measure the circumference of the circle shown in Figure 1? One way is to carefully lay a string along the line, mark the ends, straighten the string, and then use a ruler to measure the distance between the marked points. Unfortunately, this process is tedious since it is hard to juggle a string into place along a curved line and, besides, one doesn’t always have a string handy. Another way is to divide the line into a series of segments each of which appears to be straight. This is illustrated in Figure 2. Then you use a ruler to measure the length of each “straight” line segment and add up the results. Next to a few of the segments we have written our estimate of their length. Why don’t you try a few? Clearly, this is
Teacher Lab Discussion
Circumference vs. Diameter
4 5 6 Intermediate & Middle Length Investigation
also tedious, especially if the circle is large, and there is always an error in replacing a true arc by an equivalent straight line no matter how small the segment. Still, we do want to find the relationship between the circumference (C) and the circle’s diameter (D), so we need a simple, child-oriented method for finding the circumference that is quick and accurate.
A nice way of finding C is shown in Figure 3. The child rolls a cylindrical object like a soft drink or orange juice can so that the object makes one full
7 mm
5 mm
4 mm
Figure 2
Figure 3
Diameter D
C, Circumference
Figure 1
C
Picture, Data Table, and Graph
With the above in mind, our picture of the experiment is shown in Figure 4. The variables D and C should be clearly labeled. Either could be the manipulated variable but one usually chooses a can based on its diameter since it is the simpler linear measurement; so D will be treated as the manipulated and C the responding variable. Ask the children what variable they think should be the manipulated. See if they come up with similar reasoning.
In our drawing we have shown three stop marks. In Question 1 we ask why. Being expert experimentalists by now the children should explain that “we need to make sure,” “we always check by taking an average,” “the can might skip,” or “I might not start at the same spot each time.” Taken together, these answers explain why we make three measurements and then average.
The data table for three well-spaced values of D is shown in Figure 5. As you go around the room ask the kids if their data “is proportional.” Because of the value of D this is not so obvious, so they will have to think about it. Look at our first two data points. D goes from 3.9 cm to 7 cm— almost, but not quite, a factor of 2. C goes from 12.5 cm to 21. cm—again almost a factor of 2. So it does look as though the data is proportional.
How to tell for sure? Graph it, of course. Then if the graph is a straight line through (0,0), the variables are proportional.
Figure 4
Figure 5
Table I
C in cm Trial 1 Trial 2 Trial 3 Average
D in cm
12.5 12.3 12.6 12. 21.2 21.6 21.6 21. 39.5 38 41 39
revolution. The distance rolled is precisely the circumference of the can. Voila! All you have to do is measure the distance rolled and you have the circumference.
As carried out in the elementary school, a few tips are in order. Try to gather rollers of three different sizes, small, medium, and large, so that you get a nice spread in diameters, which is our manipulated variable. Small should be about 3 to 4 cm (empty spools of sewing thread are perfect); medium about 6 to 8 cm (soft drink and orange juice cans are great); large, about 12 to 14 cm (a masking tape roll or coffee can will do) in diameter. Mark one end as shown in Figure 3. Have the children lay out and tape a piece of paper on the desk and draw a starting line on the paper. Then have them pick one small, one medium, and one large roller to use in the experiment. Have them mark the point where the cylinder has gone full circle. Since the start and stop lines are on the paper, they can easily measure the circumference, and you can check to see if there are any errors.
the curve or using p, then solve for the unknown quantity.
In Question 8 we want to find the diameter of a circle if C = 120 cm. Then, properly set up we have
C
D
3.14 cm 1cm
120 cm D symbols from data
D = 120 cm 1 cm 3.14 cm
× = 38.21 cm
If you do not want the children to use decimals, then round 3.14 off to 3! Although you lose accuracy (~5%), you call on the children’s ability to multiply and divide in their heads or to use the calculators without decimals. But this is a wonderful chance for the children to use their calculators and to handle decimals.
In Question 9 we look at a typical car tire with an inner diameter of 43 cm and an outer diameter of 71 cm. Solving for the inner circumference
C
D
3.14 cm 1 cm
C
43 cm
C = 43 cm
3.14 cm 1 cm
∴ × = 135.02 cm
The outer circumference is
C = 71 cm 3.14 cm 1 cm
× = 222.94 cm
How far will the tire roll in one turn? In one turn it rolls its outer circumference or 222.94 cm.
As a nice addition to the problem, have the children measure the inner and outer diameters of the tires on their family car and on a neighbor’s car and see
which one rolls the farthest on one turn.
Besides being a practical problem, Question 10 presents an interesting challenge to the children. A large hole has a circumference of 20 m, not 20 cm. What is D? New dimensions should not confuse them. Scaling up to meters we have
C
D
3.14 cm 1 cm
20 m D
D = 20 m
1 cm 3.14 cm = 6.37 m
∴ ×
The units all cancel properly!
There is a famous picture taken in the 1920’s of a group of 20 adults hugging a giant sequoia tree in California. Let’s use that information in Question 11 to find the tree’s diameter. This is a more open- ended question so let the children come up with a solution. First they will have to estimate the arm span of a typical adult. Since an adult is around 2 meters tall, and from arm span ≅ height, we can estimate that arm span ~ 2 m.
Therefore, the circumference of the tree is ~ 40 meters. Using proportional reasoning
C
D
3.14 cm 1 cm
40 m D
D =
40 m 3.14 cm
1 cm
~ 13 meters
∴ ×
Most of the children ride bikes, so Question 12 is a chance for them to do a little practical pedaling. A bicycle wheel has a diameter of 64 cm. If you go 3000 meters (~2mi) how many turns does the wheel make. It’s a two-step problem. First, the children must find the circumference:
C
D
3.14 cm 1 cm
C
64 cm
∴C = 200.96 cm
Next they determine how many turns it will make in 3000 m. First we change C into m so that
C ~ 2.01 m
where we properly round off! Then one sets up the ratio of number of turns to distance traveled:
N
D
1 turn 2.01 m
N
3000 m
N =
3000 m 2.01 m
1492 turns
turns traveled
turns
turns to store =
Does it make any difference what the circle is made of? Since all kinds of objects were used in the experiment, the answer is no. Therefore, in Question 13 , the answer is both cylinders would have the same circumference since they have the same diameter.
We conclude with a series of short problems from a wide variety of “practical” situations. You might want to treat some of these as homework or save one for a later exam question.
If they are going to make a profit, pizza manufacturers better know how to answer Question 14. For a 12-inch pizza,
C
D
3.14 inches 1 inch
C
12 inches
C =
12 inches 3.14 inches 1 inch
= 37.68 inches
×
where we have used inches all the way, which is the American way!
Question 15 is for future restaurant owners and is a two-step logic problem. First we find the circumference of the table (in feet):
C
D
1 ft
C
8 ft
C =
8 ft 3.14 ft 1 ft
= 25.12 ft
×
Now we find the number of people. Allowing 3 feet per person
N
C
1 person 3 feet
N
25.12 feet
N = 25.12 feet
1 person 3 feet
= 8.37 people
people
∴ ×
Notice units of feet cancel. We can’t have a fraction of a person, so the table will hold 8 people.
In most countries, a !s-mile race is ~800 m. If we want to build a track that covers that distance once around, then in Question 16
C
D
3.14 m 1 m
800 m D
Solving for D,
D = 800 m
1 m 3.14 m = 255 m
×
If the children can do these problems, then they not only understand circumference vs. diameter, but they are also on their way to a higher level of formal thinking including ratios and algebra. Go for it!
Materials per Team
- 1 meterstick
- 3 cylinders: small, medium, large
