Lesson #2 - Augsburg College

Geometry: A Shapely Approach to Math
Also, note that the diameter is the longest chord in a circle. Thus the longest line connecting two points on a
circle always goes through the center of the circle!
At this point, ask the kids if it is possible to draw a straight line that only touches a circle at one point. (Yes,
it is possible!) This line is called a tangent line to a circle.
Now ask your students if it is possible to draw a line through three points on the circle? (This is imossible
if the line is straight, but if it’s curved it can cross the circle at infinitely many points!) Have your kids explain
their responses.
Now, to solidify these ideas, have the kids draw a few circles of different radii with a compass. Then, for
each circle, ask them to measure the radius, diameter, and circumference. For each circle, they should write r
=, d =, and c = and then fill in the blanks after each letter. Encourage them to measure as carefully as possible.
They can use a string to measure the circumference by laying the string around the circle on the paper and then
measuring the length of the string with a ruler. This will teach them that each circle has a different radius, diameter,
and circumference, and thus that the letters r, d, and c stand for different numbers when they correspond
with different circles. Also ask them to draw a tangent line for each circle.
Ask the students if they can draw two cirlces, both with a radius of 3 but with two different circumferences?
This is impossible, as they will figure out. It is also impossible to draw two circles with the same circumference
but different radii. This is because, as the radius of a circle increases, so does its circumference, and it turns out
that the circumference and radius have a special relationship. When you divide the circumference of some circle
by the radius of the same circle, you will always get the same answer no matter how big or how small your
circle! The answer you get is pi (pronounced like apple pie). The mathematical symbol used for pi is π, and its
numerical value to five decimal places is 3.14159. Thus, using only symbols and letters, we can write π = c / d.
Before you explain all this to your students, help them to discover it on their own! For each circle that they
drew earlier, ask them to calculate (with a calculator) the value of the circumference divided by the diameter.
When they are done, they should compare the values they have calculated. Then, they can also compare their
values with the other students in the group. Although their values will not be perfectly precise due to inevitably
imperfect measuring, all their values should be close to 3.14.
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