Lesson: Circumference and Area
Introducing the Concept
Your students know how to find the area of common figures like rectangles, parallelograms, and triangles. Now they will extend their knowledge to finding the area of a circle. Spend some time helping them understand the number pi as the ratio of the circumference of a circle to its diameter. This will help them feel more comfortable with the formula for the circumference of a circle. It will also help them relate pi to the area of a circle.
Materials: 5 circular objects of different sizes, such as jar lids, for every two students; string, rulers, and blank paper for all students
Preparation: Distribute a set of jar lids to student pairs. Also distribute string, rulers, and blank paper to each student. Have students create a table on their paper similar to the one described and illustrated below.
Prerequisite Skills: Students should be able to use a ruler to measure distances.
Draw a picture of a circle on the board or overhead projector and review the definitions for circle, diameter, and radius. Introduce the concept of circumference as perimeter.
- Ask:Does anyone know what this figure is? (circle)
Draw a diameter in the circle.
Does anyone know what we call this line that passes through the center of a circle? (diameter)
What do we call the line segment from the center to a point on the circle? (radius)
Does anyone remember what we call the distance around the circle? (Circumference; if students don't know this, tell them.) - Say:Today we are going to find a way to calculate the circumference when we know the diameter or radius.
- Have students work in pairs. Show them how to find the circumference by using a piece of string and a ruler.
- Say:To find the circumference of this jar lid, we can wrap a string around it like this. (Demonstrate the process.) The length of that string is the circumference. Next, we take the string and lay it alongside a ruler to find its length. To find the diameter of the lid, we measure across the circle so the ruler passes through the center. I'd like you to do this for the five jar lids you have at your desks. Then write this information in a table like the one on the board (overhead). Finally, calculate the ratio of the circumference to the diameter and write that in your table as well.
- Say:After you have collected all the information, examine it and note anything you discover about the relationship between circumference and diameter.
Students should discover that the ratio is a little greater than 3 for each circle. The measurements will not be exact, but they should be close enough to arrive at a value of 3.1. If some students come up with a ratio not close to 3, they should recheck their measurements and their division with their partner. - Ask:What relationships did you discover about the data you collected?
Some students will say that as the circumference increased, so did the diameter. Elicit responses from students suggesting that the ratios are all about the same, something close to 3. - Say:That's right. The ratios should all be about the same. In fact, mathematicians have been able to prove that they all equal a number called pi. Pi is approximately equal to 3.14. This leads us to a formula for the circumference, which is C =
d or C = 3.14 x d. - Ask:Who can tell me how to find the circumference of a circle with a diameter of 8 m?
Students should suggest substituting 8 for d in the formula C = 3.14 x d. Have a volunteer do this on the board for the class to see. Emphasize the importance of labeling the answer. - Then have students solve the problems below.
