United States of AmericaPA

Investigation: What is pi?

Lesson

Objective

To understand the relationship between the circumference and the diameter of a circle.

Vocabulary

When discussing this activity with your classmates, be sure to use these terms.  The diagram below provides an example of these terms.

1. Circumference - The perimeter of a circle.  This would be the length of the circle if it were opened up and straightened out to a line segment.
2. Center of a Circle - The center of a circle is the point equidistant from the points on the edge of the circle.
3. Radius - The distance from the center of the circle to any point on its circumference.
4. Diameter - Any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.  (The diameter of a circle is twice the length of its radius).

Activity

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.

1. Which do you think will be longer - the height of the can or the length of the circumference of its circular base?  Record your hypothesis and discuss your guess with a partner.

1. Use the applet below to compare the length of the circumference of a circle to the length of its diameter in $5$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 $\frac{circumference}{diameter}$circumferencediameter
(1)
(2)
(3)
(4)
(5)

To gather the data for your table, you will:

• Change the diameter of the circle using the blue dot on the circle
• Then slide the slider to roll the circle out along the line.
• Once the circle is unraveled you can lay the diameter segments end to end to compare the length of the circumference to the diameter of each circle
1. Round the answers that you got when you found the ratio of $\frac{circumference}{diameter}$circumferencediameter for each of the circles to the nearest hundredth.  Compare your results to the results of your partner.  What do you notice?

2. Now, attempt to create a circle with a circumference as close as you can get to $22$22.  What is the diameter of that circle?  Round the circumference and the diameter to the nearest whole number and express the ratio $circumference:diameter$circumference:diameter as a fraction.  Compare your results to the results of your partner.  What do you notice?

3. From the last column in the table, you should have found that the length of the circumference divided by the diameter was always the same constant.  We call this constant $\pi$π.  (This is a Greek letter and is pronounced $Pi$Pi, like the dessert)

The number $\pi$π is irrational (and so the decimals go on and on forever, without ever repeating), that's why we use the symbol $\pi$π.  If a number is irrational, then we cannot express it as the ratio of two whole numbers.  Because $\pi$π is irrational, any calculation we do on a calculator with $\pi$π 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 $\pi$π.

Remember!
$\pi$π is a special number that represents the relationship between any circle's circumference and its diameter.

It doesn't matter what the diameter is - the relationship between the circumference and diameter is always the same.

$\pi=\frac{C}{d}$π=Cd

Your answer to question 3 above, is the value that is often used to approximate $\pi$π as a decimal.

Your answer to question 4 above, is the value that is often used to approximate $\pi$π as a fraction.

1. Explain in your own words what $\pi$π is and how you would approximate it as a decimal number or as a fractional number.  Discuss your explanation of what $\pi$π is to your partner.

2. Go back to question 1.  Do you still agree with your hypothesis about the tennis ball can?  Why or why not?  Discuss your reasoning with your partner.

Outcomes

CC.2.3.7.A.1

Solve real-world and mathematical problems involving angle measure, area, surface area, circumference, and volume.

M07.C-G.2.2.1

Find the area and circumference of a circle. Solve problems involving area and circumference of a circle(s). Formulas will be provided.
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