Lesson 3
Exploring Circumference
Lesson Narrative
In this lesson, students discover that there is a proportional relationship between the diameter and circumference of a circle. They use their knowledge from the previous unit on proportionality to estimate the constant of proportionality. Then they use the constant to compute the diameter given the circumference (and vice versa) for different circles. We define \(\pi\) as the value of the constant and discuss various commonly used approximations. In the next lesson, students use various approximations for pi to do computations. Also, relating the circumference to the radius is saved for the next lesson.
Determining that the relationship between the circumference and diameter of circles is proportional is an example of looking for and making use of structure (MP7).
Learning Goals
Teacher Facing
 Comprehend the word “pi” and the symbol $\pi$ to refer to the constant of proportionality between the diameter and circumference of a circle.
 Create and describe (in writing) graphs that show measurements of circles.
 Generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.
Student Facing
Let’s explore the circumference of circles.
Required Materials
Required Preparation
Household items: collect circular or cylindrical objects of different sizes, with diameters from 3 cm to 25 cm. Each group needs 3 items of relatively different sizes. Examples include food cans, hockey pucks, paper towel tubes, paper plates, CD’s. Record the diameter and circumference of the objects for your reference during student work time.
The empty toilet paper roll is for optional use during the warmup as a demonstration tool.
You will need one measuring tape per group of 24 students. Alternatively, you could use rulers and string.
Learning Targets
Student Facing
 I can describe the relationship between circumference and diameter of any circle.
 I can explain what $\pi$ means.
CCSS Standards
Glossary Entries

circle
A circle is made out of all the points that are the same distance from a given point.
For example, every point on this circle is 5 cm away from point \(A\), which is the center of the circle.

circumference
The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius \(r\) then the circumference is \(2\pi r\).
The circumference of a circle of radius 3 is \(2 \boldcdot \pi \boldcdot 3\), which is \(6\pi\), or about 18.85.

diameter
A diameter is a line segment that goes from one edge of a circle to the other and passes through the center. A diameter can go in any direction. Every diameter of the circle is the same length. We also use the word diameter to mean the length of this segment.

pi ($\pi$)
There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is \(\pi\).
We can represent this relationship with the equation \(C=\pi d\), where \(C\) represents the circumference and \(d\) represents the diameter.
Some approximations for \(\pi\) are \(\frac{22}{7}\), 3.14, and 3.14159.

radius
A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.
For example, \(r\) is the radius of this circle with center \(O\).