This article describes a math activity for learning about circumference.
CCSS.MATH.CONTENT.7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Circles have several properties, including diameter, radius, area, and circumference. The diameter, radius and area are often described in expressions and equations that are related to π ("pi").
The radius is the segment from the center to the perimeter of the circle. The diameter is twice the the length of the radius. The diameter is the width of the circle, through its center, which makes it a very useful measurement. The area is often calculated in terms of the radius or diameter.
However, the circumference is almost never discussed with as much detail. This activity provides an opportunity for the student to become more familiar with the practical application of circumference and how it relates to pi.
A paper towel roll is helpful in a kitchen. This prior knowledge can help students build new ideas about geometry topics related objects with round surfaces. Daily interactions with common household objects, such as paper towel rolls, can strengthen the relevance of math in everyday life.
From observation, we see that a circumference is the perimeter around a circle. If you walk around the entire perimeter of a circle, we find that the total distance around the circle is equal to the length of a segment that has been "un-rolled" from a round arc into a straight line.
Let us take a cardboard paper towel roll. The cross section is a circle.
If we make a single cut along its longest side, we can "un-roll" the paper towel roll into a flat sheet of cardboard. (The straightness of the cut does not need to be perfect.)
We know that a circumference is the length around a circle. We also know that the length remains unchanged. Also, the surface area of the paper towel roll along its longest length is also unchanged. The only change was the path of the length, which was transformed from a round arc into a straight line. This activity demonstrates the powerful the idea of π ("pi").
Pi is an important math tool that allows us to relate the diameter of an object to its circumference without the need for tedious physical measurement or destructive modification.
We also see in this activity that circumference is simply the arc length that travels completely around a circle.