Remembering the formulae for area and circumference of a circle is often a challenge for students. I like to tackle them one at a time and give students a chance to explore the origin of each formula in this inductive task. This works to my students’ strengths as inductive thinkers and reinforces a conceptual understanding of mathematics.
Let’s look at circumference today.
Supplies (Per Group)
You’ll need:
- String, long enough to go around any of the images
- A ruler
- Paper for tracking data
- Printouts of several famous circles.
I used:
- Columbus Circle, in New York
- Place Charles de Gaulle, Paris
- The circle around Stonehenge
- The circle around Mickey Mouse & Walt Disney, Disneyland
- Copernicus Lunar Crater, The Moon, for more info
- Not so famous, but my Alma Mater: Ring Road, UC Irvine
Begin by demonstrating
Take the photo of Columbus Circle and measure the distance across the circle using the ruler. Explain that you’re walking straight across the widest part of the circle. Introduce this as the diameter and emphasize that it represents the distance of walking across the entire circle.
Then, determine how far it would be to walk around the circle (note that this is called the circumference). Curve the string around the circle and mark the distance. Straighten the string out and measure using the ruler.
Jot this measurement down in a table.
| Circle | Across (D) | Around (C) |
|---|---|---|
| Columbus | 7″ | 22″ |
| Charles de Gaulle | ||
| Stonehenge | ||
| Disneyland |
Ask for any observations. Students might point out that it’s much further to go around than to go across. Someone might note it’s about three times the length.
Students Measure
Now ask students to find the diameter and circumference for the remaining four images and note the information in their table. Set up an order or give specific jobs to avoid arguments and encourage collaboration.
Explore The Relationships
Once all the measurements are complete, fill in your own chart by calling on groups. Try to get a class consensus about the measurements.
Now explain that it is possible to find the length around the circle by knowing only the distance across the circle. There is a mathematical relationship between these two measurements that will work for any circle. Add a new column to the end of the table and ask students to determine the relationship between D and C.
| Circle | Across (D) | Around (C) | Ratio (C รท D) |
|---|---|---|---|
| Columbus | 7″ | 22″ | ~3 |
| Charles de Gaulle | |||
| Stonehenge | |||
| Disneyland |
Discuss Pi
Once the groups are finished, ask your class to examine the pattern arising in this new column. They should make a generalization about the relationship between C and D. Discuss this.
Students should come to the conclusion that the circumference, or distance around, is about three times longer than the diameter, or distance across.
Explain that they are discovering a mathematical fact that people have been investigating for 4,000 years. They’ve come close to finding a very strange number, called ฯ, which has no known end. We commonly abbreviate it as 3.14 .
It is exactly ฯ times longer to go around a circle than to go across a circle. In other words, the circumference is equal to ฯ times the diameter or c = ฯd.
Naturally, you can choose to go into ฯ however deeply you have time for (and your students have interest in), but, presented well, ฯ is something that fascinates a good number of kids.
Practice With Radius
Also ask the question “What if we only know half the distance across the circle? Can we still find the circumference?” Emphasize this question using one of the photos. Discuss with students that you would simply double the distance, and then multiply by ฯ.
Explain that walking from the edge of a circle to the center, or half the diameter, has the special name of radius. Naturally, two times the radius is the same as the diameter or d = 2r.
Reminders
When practicing later, always refer back to the idea that walking around Stonehenge is exactly ฯ times longer than walking across it. This conceptual understanding should help combat confusion, especially when you introduce the area of a circle.