## The Ratio of Circumference to Diameter

• Lesson
6-8
2

Students measure the circumference and diameter of circular objects. They calculate the ratio of circumference to diameter for each object in an attempt to identify the value of pi and the circumference formula.

Prior to this lesson, ask students to bring in several flat, circular objects that they can measure.

As a warm-up, ask students to measure the length and width of their desktops. Ask them to decide which type of unit should be used. Then, have students measure or calculate the distance around the outside of their desktops.

With the class, discuss the following questions:

1. What unit did you use to measure your desks? Why?
[Because of the size of desks, the most appropriate units are probably inches or centimeters.]
2. Why did some of your classmates get different measurements for the dimensions of their desks?
[Measurements will obviously differ because of the units. In addition, the level of precision may give different results. For instance, a student may round to the nearest inch, while another may approximate to the nearest ¼-inch.]
3. What do we call the distance around the outside of an object?
[The distance around the outside of a polygon is known as the perimeter. The distance around the outside of a circle is known as the circumference.]

Inform the class that they will be measuring the circumference of several circular objects during today’s lesson. Also, alert them that, just as there is a formula for finding the perimeter of a rectangle (P = 2L + 2W), there is also a formula for finding the circumference of a circle. They should keep their eyes open for a formula as they proceed through the measurement activities.

Divide the class into groups of four students. Within the groups, each student will be given a different job. (If class size is not conducive to four students per group, form groups of three — one student can be assigned two jobs.)

• Task Leader: Ensures all students are participating; lets the teacher know if the group needs help or has a question.
• Recorder: Keeps group copy of measurements and calculations from activity.
• Measurer: Measures items (although all students should check measurements to ensure accuracy).
• Presenter: Presents the group’s findings and ideas to the class.

Students should measure the "distance around" and the "distance across" of the objects that they brought to school. Students will likely have little trouble measuring the distance across, although they may have some difficulty identifying the exact middle of an object. To measure the distance around, students will likely need some assistance. An effective method for measuring the circumference is to wrap a string around the object and then measure the string. To ensure accuracy, care should be taken to keep the string taut when measuring the outside of a circular object.

Students should be allowed to select which unit of measurement to use. However, instruct students to use the same unit for the distance around and the distance across.

Students should record the following information in the Apple Pi activity sheet:

• Description of each object
• Distance around the outside of each object
• Distance across the middle of each object
• Distance around divided by distance across
 Apple Pi Activity Sheet

After the measurements have been recorded, a calculator can be used to divide the distance around by the distance across. Students should answer both questions on the worksheet. As students are working, take note of their results. Push students to note any numbers in the last column that seem to be irregular, and have them check their measurements for these rows.

When all groups have completed the measurements and calculations, conduct a whole-class discussion. Rather than present each individual object, students should discuss the average and note any interesting findings. Students should also compare their averages with those of other groups.

You may wish to use the Circle Tool applet as a demonstration tool. This applet allows students to see many other circles of various sizes, as well as the corresponding ratio of circumference to diameter.

 Circle Tool

Explain that each group has found an approximation for the ratio of the distance around to the distance across, and this ratio has a special name: π. (It may also be necessary to explain that the "distance across" is known as the diameter and that the "distance around" is known as the circumference. Because of this relationship, algebraic notation can be used to write

circumference ÷ diameter = π

or, said another way,

π = C/d

which leads to the following formula for circumference:

C = π × d

Point out that groups within the class may have obtained slightly different approximations for π. Explain that determining the exact value of π is very hard to calculate, so approximations are often used. Discuss various approximations of π that are acceptable in your school’s curriculum.

• Pieces of string, approximately 48" long
• Circular objects to be measured
• Apple pies (or other circular food item, to be measured at the end of the lesson)
• Apple Pi Activity Sheet
• Calculators
• Rulers

Assessments

1. Each group can be given an apple pie (or other acceptable substitute) and will find its circumference by measuring the diameter and using the formula.

2. Students should practice using the formula C = π × d as independent work. Their work on such problems could be used for assessment. Two real world problems are:

• According to Guinness, the world’s largest rice cake measured 5.83 feet in diameter. What is the circumference of this rice cake?
• The tallest tree in the world is believed to be the Mendicino Tree, a redwood near Ukiah, California, that is 112 meters tall! Near the ground, the circumference of this tree is about 9.85 meters. The age of a redwood can be estimated by comparing its diameter to trees with similar diameters. What is the diameter of the Mendicino Tree?
Extensions
1. In this lesson, students use a numeric approach to see the relationship between circumference and diameter. That is, students compute the ratio of circumference to diameter and then take the average for several objects. For a visual approach, have students plot the diameter of those objects along the horizontal axis of a graph and plot the circumference along the vertical axis. As shown below, a line of best fit with slope of roughly 3.14, or π, will approximate the points in the resulting scatterplot.

2. Students can read and react to the book Sir Cumference and the First Round Table: A Math Adventure by Cindy Neuschwander. Within their groups, students can pose questions about the book and its mathematical accuracy, realism, and other components.
3. In their groups, students can research the history of π and its calculation, approximation, and uses. In particular, they can research Archimedes method for estimating the area of a circle using inscribed polygons. The students could report their findings to the class.

Questions for Students

1. Why did we use the ratio of circumference to diameter for several objects? Wouldn’t we have gotten the same result using just one object?

[If we had used just one object, an incorrect measurement would have given an incorrect approximation for π. Using several objects ensures that our results are correct. In addition, slight errors in measurement may give different values of π, so using the average of several measurements will help to eliminate rounding errors.]

2. Were any of the ratios in the last column not close to 3.14? If not, explain what might have happened.

[The ratio of circumference to diameter is always the same, and the ratio is always close to 3.14. If a value in the last column is not close to 3.14, it is the result of a measurement or calculation error.]

3. Describe some situations in which knowing the circumference (and how to calculate it) would be useful.

[Bike tires are often described by their diameter. For instance, a 26-inch tire is a tire such that the diameter is 26". Each time the tire makes one complete rotation, the bike moves forward a distance equal to the circumference of the tire. Therefore, it would be helpful to know how to calculate the circumference based on the diameter.]

Teacher Reflection

• What prior knowledge did students have of π (if any)? How did student’s prior knowledge affect the delivery of the lesson? What modifications did you need to make as a result, and how effective were these adjustments?
• How precise were student measurements? How did you assist students with their measurements?
• How did students react to the use of 3.14 as an approximation of π? Were there any adverse reactions due to conceptual misunderstandings?
• How did students show that they were actively learning?
• Did students understand that the ratio of circumference to diameter (i.e., π) is an approximation? Did they understand why they had obtained different values for this approximation during the activity?

### Discovering the Area Formula for Circles

6-8
Using a circle that has been divided into congruent sectors, students will discover the area formula by using their knowledge of parallelograms. Students will then calculate the area of various flat circular objects that they have brought to school. Finally, students will investigate various strategies for estimating the area of circles.

### Learning Objectives

Students will:

• Measure the circumference and diameter of various circular objects
• Calculate the ratio of circumference to diameter
• Discover the formula for the circumference of a circle