3.1.5: Circumference and Wheels

Lesson

Let's explore how far different wheels roll.

Exercise \(\PageIndex{1}\): A Rope and a Wheel

Han says that you can wrap a 5-foot rope around a wheel with a 2-foot diameter because \(\frac{5}{2}\) is less than pi. Do you agree with Han? Explain your reasoning.

Exercise \(\PageIndex{2}\): Rolling, Rolling, Rolling

Your teacher will give you a circular object.

1. Follow these instructions to create the drawing:
   • On a separate piece of paper, use a ruler to draw a line all the way across the page.
   • Roll your object along the line and mark where it completes one rotation.
   • Use your object to draw tick marks along the line that are spaced as far apart as the diameter of your object.
2. What do you notice?
3. Use your ruler to measure each distance. Record these values in the first row of the table:
   1. the diameter of your object
   2. how far your object rolled in one complete rotation
   3. the quotient of how far your object rolled divided by the diameter of your object

      Table \(\PageIndex{1}\)

      object	diameter	distance traveled in one rotation	distance \(\div\) diameter

4. If you wanted to trace two complete rotations of your object, how long of a line would you need?
5. Share your results with your group and record their measurements in the table.
6. If each person in your group rolled their object along the entire length of the classroom, which object would complete the most rotations? Explain or show your reasoning.

Exercise \(\PageIndex{3}\): Rotations and Distance

1. A car wheel has a diameter of 20.8 inches.
   1. About how far does the car wheel travel in 1 rotation? 5 rotations? 30 rotations?
   2. Write an equation relating the distance the car travels in inches, \(c\), to the number of wheel rotations, \(x\).
   3. About how many rotations does the car wheel make when the car travels 1 mile? Explain or show your reasoning.
2. A bike wheel has a radius of 13 inches.
   1. About how far does the bike wheel travel in 1 rotation? 5 rotations? 30 rotations?
   2. Write an equation relating the distance the bike travels in inches, \(b\), to the number of wheel rotations, \(x\).
   3. About how many rotations does the bike wheel make when the bike travels 1 mile? Explain or show your reasoning.

Are you ready for more?

Here are some photos of a spring toy.

Figure \(\PageIndex{1}\): 4 images of a slinky. Image top left is a compressed slinky. Image top right is slinky with both ends on the floor. Image bottom left is measuring the diameter. Image bottom right is the slinky stretched out.

If you could stretch out the spring completely straight, how long would it be? Explain or show your reasoning.

Exercise \(\PageIndex{4}\): Rotations and Speed

The circumference of a car wheel is about 65 inches.

1. If the car wheel rotates once per second, how far does the car travel in one minute?
2. If the car wheel rotates once per second, about how many miles does the car travel in one hour?
3. If the car wheel rotates 5 times per second, about how many miles does the car travel in one hour?
4. If the car is traveling 65 miles per hour, about how many times per second does the wheel rotate?

Summary

The circumference of a circle is the distance around the circle. This is also how far the circle rolls on flat ground in one rotation. For example, a bicycle wheel with a diameter of 24 inches has a circumference of \(24\pi\) inches and will roll \(24\pi\) inches (or \(2\pi\) feet) in one complete rotation. There is an equation relating the number of rotations of the wheel to the distance it has traveled. To see why, let’s look at a table showing how far the bike travels when the wheel makes different numbers of rotations.

Table \(\PageIndex{2}\)

number of rotations	distance traveled (feet)
\(1\)	\(2\pi\)
\(2\)	\(4\pi\)
\(3\)	\(6\pi\)
\(10\)	\(20\pi\)
\(50\)	\(100\pi\)
\(x\)	\(?\)

In the table, we see that the relationship between the distance traveled and the number of wheel rotations is a proportional relationship. The constant of proportionality is \(2\pi\).

To find the missing value in the last row of the table, note that each rotation of the wheel contributes \(2\pi\) feet of distance traveled. So after \(x\) rotations the bike will travel \(2\pi x\) feet. If \(d\) is the distance, in feet, traveled when this wheel makes \(x\) rotations, we have the relationship: \(d=2\pi x\).

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 \(\PageIndex{2}\)

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 \(\PageIndex{3}\)

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 \(\PageIndex{4}\)

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 \(\PageIndex{5}\)

Practice

Exercise \(\PageIndex{5}\)

The diameter of a bike wheel is 27 inches. If the wheel makes 15 complete rotations, how far does the bike travel?

Exercise \(\PageIndex{6}\)

The wheels on Kiran's bike are 64 inches in circumference. How many times do the wheels rotate if Kiran rides 300 yards?

Exercise \(\PageIndex{7}\)

The numbers are measurements of radius, diameter, and circumference of circles A and B. Circle A is smaller than circle B. Which number belongs to which quantity?

\(2.6, 6, 7.6, 15.2, 15.7, 47.7\)

(From Unit 3.1.4)

Exercise \(\PageIndex{8}\)

Circle A has circumference \(2\frac{2}{3}\) m. Circle B has a diameter that is \(1\frac{1}{2}\) times as long as Circle A’s diameter. What is the circumference of Circle B?

(From Unit 3.1.3)

Exercise \(\PageIndex{9}\)

The length of segment \(AE\) is 5 centimeters.

Figure \(\PageIndex{6}\)

1. What is the length of segment \(CD\)?
2. What is the length of segment \(AB\)?
3. Name a segment that has the same length as segment \(AB\)?

(From Unit 3.1.2)