To understand the relationship between the circumference and the diameter of a circle.
When discussing this activity with your classmates, be sure to use these terms. The diagram below provides an example of these terms.
Consider the following question and without measuring, make a hypothesis about what the correct answer will be. The can in the image below contains three tennis balls.
The height of the can is equal to the length of the diameter of the three tennis balls added together. The circumference of the base of the can is the same as the circumference of the circle formed by slicing the tennis ball through its center.
Use the applet below to compare the length of the circumference of a circle to the length of its diameter in 5 different circles. Record your results for the length of the circumference, the length of the diameter, and the result you get when you divide the circumference by the diameter, in the table below.
circle | length of diameter | length of circumference | \frac{circumference}{diameter} |
---|---|---|---|
(1) | |||
(2) | |||
(3) | |||
(4) | |||
(5) |
To gather the data for your table, you will:
The number \pi is irrational (and so the decimals go on and on forever, without ever repeating), that's why we use the symbol \pi. If a number is irrational, then we cannot express it as the ratio of two whole numbers. Because \pi is irrational, any calculation we do on a calculator with \pi in its decimal form will be an estimate. A pretty close one, but still not what we would call exact. When we want to write an expression in "exact form", we can write it in terms of some multiple of \pi.
It doesn't matter what the diameter is - the relationship between the circumference and diameter is always the same.
\pi = \frac{C}{d}
Your answer to question 3 above, is the value that is often used to approximate \pi as a decimal.
Your answer to question 4 above, is the value that is often used to approximate \pi as a fraction.
Given the formulas for the area and circumference of a circle, use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.