3.1.3: Exploring Circumference

Lesson

Let's explore the circumference of circles.

Exercise \(\PageIndex{1}\): Which is Greater?

Clare wonders if the height of the toilet paper tube or the distance around the tube is greater. What information would she need in order to solve the problem? How could she find this out?

Figure \(\PageIndex{1}\): Three different views of a circular shaped toilet paper tube. The first view is of the vertical height of the tube. The second view is of the circular base of the tube. The third view is of both the base and the height of the tube.

Exercise \(\PageIndex{2}\): Measuring Circumference and Diameter

Your teacher will give you several circular objects.

1. Explore the applet to find the diameter and the circumference of three circular objects to the nearest tenth of a unit. Record your measurements in the table.
2. Plot the diameter and circumference values from the table on the coordinate plane. What do you notice?
3. Plot the points from two other groups on the same coordinate plane. Do you see the same pattern that you noticed earlier?

Exercise \(\PageIndex{3}\): Calculating Circumference and Diameter

Here are five circles. One measurement for each circle is given in the table.

Figure \(\PageIndex{2}\)

Use the constant of proportionality estimated in the previous activity to complete the table.

Table \(\PageIndex{1}\)

	diameter (cm)	circumference (cm)
circle A	\(3\)
circle B	\(10\)
circle C		\(24\)
circle D		\(18\)
circle E	\(1\)

Are you ready for more?

The circumference of Earth is approximately 40,000 km. If you made a circle of wire around the globe, that is only 10 meters (0.01 km) longer than the circumference of the globe, could a flea, a mouse, or even a person creep under it?

Summary

There is a proportional relationship between the diameter and circumference of any circle. That means that if we write \(C\) for circumference and \(d\) for diameter, we know that \(C=kd\), where \(k\) is the constant of proportionality.

The exact value for the constant of proportionality is called \(\pi\). Some frequently used approximations for \(\pi\) are \(\frac{22}{7}\), 3.14, and 3.14159, but none of these is exactly \(\pi\).

Figure \(\PageIndex{3}\)

We can use this to estimate the circumference if we know the diameter, and vice versa. For example, using 3.1 as an approximation for \(\pi\), if a circle has a diameter of 4 cm, then the circumference is about \((3.1)\cdot 4=12.4\) or 12.4 cm.

The relationship between the circumference and the diameter can be written as

\(C=\pi d\)

Glossary Entries

Definition: 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.

Figure \(\PageIndex{4}\)

Definition: 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 \(2\cdot\pi\cdot 3\), which is \(6\pi\), or about 18.85.

Definition: 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.

Figure \(\PageIndex{5}\)

Definition: 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.

Figure \(\PageIndex{6}\)

Definition: 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 \(O\).

Figure \(\PageIndex{7}\)

Practice

Exercise \(\PageIndex{4}\)

Diego measured the diameter and circumference of several circular objects and recorded his measurements in the table.

Table \(\PageIndex{2}\)

object	diameter (cm)	circumference (cm)
half dollar coin	\(3\)	\(10\)
flying disc	\(23\)	\(28\)
jar lid	\(8\)	\(25\)
flower pot	\(15\)	\(48\)

One of his measurements is inaccurate. Which measurement is it? Explain how you know.

Exercise \(\PageIndex{5}\)

Complete the table. Use one of the approximate values for \(\pi\) discussed in class (for example 3.14, \(\frac{22}{7}\), 3.1416). Explain or show your reasoning.

Table \(\PageIndex{3}\)

object	diameter	circumference
hula hoop	\(35\) in
circular pond		\(556\) ft
magnifying glass	\(5.2\) cm
car tire		\(71.6\) in

Exercise \(\PageIndex{6}\)

\(A\) is the center of the circle, and the length of is 15 centimeters.

Figure \(\PageIndex{8}\)

1. Name a segment that is a radius. How long is it?
2. Name a segment that is a diameter. How long is it?

(From Unit 3.1.2)

Exercise \(\PageIndex{7}\)

1. Consider the equation \(y=1.5x+2\). Find four pairs of \(x\) and \(y\) values that make the equation true. Plot the points \((x,y)\) on the coordinate plane.

Figure \(\PageIndex{9}\)

2. Based on the graph, can this be a proportional relationship? Why or why not?

(From Unit 2.4.1)