Properties of Circles
(Teacher Copy)
Activity 1: Finding p
Students measure circular objects to discover that the ratio of the
circumference to the radius of a circle is p.
To find relationships between the radius, diameter, and circumference
of a circle.
Introduction:
Ask the students if they can tell or show you the meaning of the terms
radius, diameter, and circumference (see the Reference
Copy). Can they think of when they might want to know the radius,
diameter, or circumference of a circle? Discuss why these concepts
are important when buying automobile tires. (Tire sizes are based
on tire radius and width. You might want to have students examine
tire ads and discuss the various sizes that are available for a given car.)
Part A
Object 
All objects 
diameter 
Answers will vary 
radius 
Answers will vary 
circumference 
Answers will vary 
diameter divided by radius 
» 2 for all entries 
circumference divided by radius 
» 6.2 for all entries 
circumference divided by diameter 
» 3.1 for all entries 
Summary Questions

Students should say something to the effect that the size of the circumference
increases as the radius (diameter) increases.

Students should notice that the answers in the last three rows of the table
are the same across the row.

Yes, you should get the same ratios for any circle.

The calculated circumferences should be close to the measured circumferences
in the table.

Students should report that calculated circumferences are close to measured
circumferences in the table.
Closing Discussion

When you divide the circumference by the radius of any circle, will you
always get about 3.1? Yes, you will always get an approximation of
p.

If you measured the radius, diameter, and circumference in inches instead
of centimeters, would it change the results of the experiment? (If
time permits, you may want to try this out!) No, the ratio is a constant.

Why do you get some numbers that are a little more or a little less than
3.1? Accuracy of measurement will affect the calculations.
Activity 2: Graphing Relationships
Students will create a graph of the diameter in relation to the circumference
of a circle. It is assumed that students have completed Activity
1 and have the student copy available. As an alternative to graph
paper, have students use computer graphing software to make plots.
To find relationships between the radius, diameter, and circumference
of a circle.
Introduction:
Ask students how they could make a graphic representation of
the numbers they found in Activity 1.
[Review with students how to plot pairs of points on an xy graph.]
The graphs drawn by students should be similar to the graph in
the Student Copy. The line
drawn should have approximately an equal number of points on each side
of the line. Numbering on the graphs should be in equal increments.
Part C:
The answer to ad is p. Student answers
should be about 3.1 but may vary due to measurement error.
Summary Questions

Students should say something to the effect that plotted measurements of
other circles would appear on or near the line.

The ratio of circumference to diameter is a constant; it is a linear relationship.
Closing Discussion

Why did we draw one straight line on the graph rather than "connect the
dots"? There is a linear relationship between diameter and circumference
of a circle; the points plotted are only approximations of this linear
relationship.

Is there a relationship between p and the slope
of a line? The slope of the line is p.
© Copyright
Area 10 Mathematics and Technology Professional Development Center
Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999
Comments: egalindo@indiana.edu
http://www.indiana.edu/~atmat/units/ratio/ratio_t1.htm
