*Title*

Changes in the Circumference

*Problem Statement*

What happens to the circumference of a circle if you double the diameter? If you triple the diameter? If you halve the diameter? As the diameter increases (or decreases) in measure,
how does the circumference change? Why
does this change occur?

*Problem setup*

How does changing the diameter of a circle affect its circumference?

*Plans to Solve/Investigate
the Problem*

Using Geometer’s Sketchpad, we look at the formula for the circumference of a circle C=2 p r or C = pd where p=3.14. By changing the measure of the diameter of the circle, we can see the calculation changes in the circumference.

*Investigation/Exploration of
the Problem*

We started with a circle (red) with a diameter equal to 1.0 inch (radius equal
to .50 inch) and found that the circumference is 3.15 inches. We next created a circle (black) with a
diameter double the first, so that radius equals 1.0 inch and diameter equals
2.0 inches. We found that the circumference
also doubled. Next we created a circle (blue)
with a diameter half the original circle, so that the diameter equals .50 and
the circumference became 1.57. Next, we created a circle (green) with a diameter three times the original, so
that the diameter equals 3.0 inches and the circumference became 9.42 inches;
again approximately triple the original circumference of 3.15 inches.

Click
here to see GSP file

The formula for the circumference of a circle is 2pr or pd. Since p is approximately 3.14, the circumference is going to be approximately three times the diameter of any circle.

*Extensions of the Problem*

What happens to the area of a circle when its original diameter is doubled? Tripled?

Click
here to see GSP file

When the
diameter of the original circle is equal to 1 inch, the area is 0.79 inches^{2}. Doubling the diameter of the circle to equal
2 inches creates an area of 3.16 inches^{2}, and tripling the diameter
to 3 inches makes the area equal to7.06 inches^{2}. Looking at the formula for the area of a
circle we find that A = pr^{2} where p is equal to
3.14. Since the radius is not doubled,
but instead is squared, the area changes according to the square of the radius
of the circle.

*Author & Contact*

Jill Jackson and Shirley Crawford

jjackson@rockdale.k12.ga.us or
scrawford@rockdalek.12.ga.us