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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?
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.
Your teacher will give you two circular objects.
Measure the diameter and the circumference of each circle to the nearest tenth of a centimeter. Record your measurements in the first two rows of the table.
| object | diameter (cm) | circumference (cm) |
|---|---|---|
Plot your diameter and circumference values on the coordinate plane. What do you notice?
A coordinate plane with the origin labeled "O". The horizontal axis is labeled "diameter, in centimeters," and the numbers 0 through 25, in increments of 5, are indicated. The vertical axis is labeled "circumference, in centimeters," and the numbers 0 through 80, in increments of 10, are indicated.
Find out the measurements from another group that measured different objects. Record their values in your table and plot them on your same coordinate plane.
What do you notice about the diameter and circumference values for these four circles?
Here are five circles. One measurement for each circle is given in the table.
Use the constant of proportionality estimated in the previous activity to complete the table.
| diameter (cm) | circumference (cm) | |
|---|---|---|
| circle A | 3 | |
| circle B | 10 | |
| circle C | 24 | |
| circle D | 18 | |
| circle E | 1 |
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, and its symbol is \(\boldsymbol\pi\). Some frequently used approximations for \(\pi\) are \(\frac{22} 7\), 3.14, and 3.14159, but none of these is exactly \(\pi\).
A graph of a line in the coordinate plane with the origin labeled O. The horizontal axis is labeled “d” and the numbers 1 through 6 are indicated. The vertical axis is labeled “C” and the numbers 2 through 12, in increments of 2, are indicated. The line begins at the origin, slants upward and to the right, and passes through the point 1 comma pi.
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 \((3.1)\boldcdot 4 = 12.4\), or 12.4 cm.
The relationship between the circumference and the diameter can be written as
\(\displaystyle C = \pi d\)
There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is
This relationship can be represented with the equation
Some approximations for