Circumference of a Circle

Task

1. Draw circles with diameters as indicated below and measure their circumferences to complete the following table.

   Diameter of Circle (inches)	Circumference of Circle (inches)	$\frac{\text{Circumference of Circle}}{\text{Diameter of Circle}}$
   1
   2
   3
   $\frac{1}{2}$

2. The number $\pi$ can be defined as the circumference of a circle with diameter 1 (unit). Using your knowledge about circles (that is, without measuring), complete the following table. Explain how you know the circumferences of the different circles.

   Diameter of Circle (inches)	Circumference of Circle (inches)	$\frac{\text{Circumference of Circle}}{\text{Diameter of Circle}}$
   1	$\pi$	$\pi$
   2
   3
   $\frac{1}{2}$

3. How does the information in the two tables compare? Explain.

IM Commentary

The goal of this task is to study the circumferences of different sized circles, both using manipulatives and from the point of view of scaling. Measuring the circumference of a circle is challenging because most rulers are designed to measure straight distances. A good technique would be to use a piece of string to wrap around the circles and then the string can be straightened out and measured. The teacher will have to supply compasses, rulers, and string for part (a) of the task.

Tape measures and rulers using inches are usually divided into halves, quarters, eighths, and sixteenths of an inch so students should record the answers in the first table as fractions. The task provides a good opportunity to discuss accuracy of measurements. In part (a) of the task, there are many sources of inaccuracy:

• The circles drawn with a compass are only approximately circular
• The string cannot be laid down exactly along the circular path
• Measuring the length of the string with a ruler only allows a certain level of accuracy

It is important for students to realize that the formula for the circumference of a circle applies to mathematical objects only, not to the approximate circles we encounter in the world around us. Because of the several issues with accuracy, it is possible that the first table will not be a ratio table: we can only expect the ratios of circle circumferences to their diameters will be approximately equivalent.

Solution

1. Answers here will differ. For the circle with diameter one inch, for example, answers will likely vary between 3 inches and 3$\frac{1}{4}$ inch and could be more (or less) depending on how careful the string is laid out around the circles. The table shows approximate answers:

    

   Diameter of Circle (inches)	Circumference of Circle (inches)	$\frac{\text{Circumference of Circle}}{\text{Diameter of Circle}}$
   1	3$\frac{1}{8}$	3$\frac{1}{8}$
   2	6$\frac{1}{4}$	3$\frac{1}{8}$
   3	9$\frac{3}{8}$	3$\frac{1}{8}$
   $\frac{1}{2}$	1$\frac{1}{2}$	3

2. To get a circle of diameter two inches from a circle of radius one inch we apply a scale factor of 2, which will double all lengths. The circumference of the circle is a measure of its ''length'' and so the circumference of a circle with diameter 2 inches will be twice the circumference of a circle with diameter 1 inch. In the same way, we apply a scale factor of 3 and $\frac{1}{2}$ to get the circles with diameter 3 inches and $\frac{1}{2}$ inch from the circle of radius 1 inch. The circumference of these circles will be scaled by the same factor as listed below in the table:

   Diameter of Circle (inches)	Circumference of Circle (inches)	$\frac{\text{Circumference of Circle}}{\text{Diameter of Circle}}$
   1	$\pi$	$\pi$
   2	2$\pi$	$\pi$
   3	3$\pi$	$\pi$
   $\frac{1}{2}$	$\frac{1}{2}\pi$	$\pi$

3. The first table uses hand-drawn models of circles while the second table is for the abstract mathematical objects modeled by the drawings. This means that the diameters and circumferences in the second table are exact while they are only approximate for the first table. The first two columns of the second table is a ratio table while the first two columns of the first table may or may not be depending on how accurate the measurements are.  

   The number $\pi$ is approximately equal to 3.1416 so an estimate of $3\frac{1}{8} = 3.25$ or $3$ is reasonable using string and a ruler.