Lesson 7Exploring the Area of a Circle

Learning Goal

Let’s investigate the areas of circles.

Learning Targets

  • I know whether or not the relationship between the diameter and area of a circle is proportional and can explain how I know.

  • If I know a circle’s radius or diameter, I can find an approximation for its area.

Lesson Terms

  • area of a circle

Warm Up: Estimating Areas

Problem 1

Your teacher will show you some figures. Decide which figure has the largest area. Be prepared to explain your reasoning.

Activity 1: Estimating Areas of Circles

Problem 1

Your teacher will assign your group two circles of different sizes.

  1. Set the diameter of your assigned circle and use the applet to help estimate the area of the circle.

    Note: to create a polygon, select the Polygon tool, and click on each vertex. End by clicking the first vertex again. For example, to draw triangle , click on ---.

    open applet in presentation mode

  2. Record the diameter in column and the corresponding area in column for your circles and others from your classmates.

    open applet in presentation mode

  3. In a previous lesson, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?

Print Version

Your teacher will give your group two circles of different sizes.

  1. For each circle, use the squares on the graph paper to measure the diameter and estimate the area of the circle. Record your measurements in the table.

    diameter (cm)

    estimated area (cm)

  2. Plot the values from the table on the class coordinate plane. Then plot the class’s data points on your coordinate plane.

    A coordinate plane with the origin labeled O. The horizontal axis is labeled "diameter, in centimeters," and the numbers 2 through 22, in increments of 2, are indicated. The vertical axis is labeled "area, in square centimeters, and the numbers 50 through 300, in increments of 50, are indicated.

  3. In a previous lesson, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?

Are you ready for more?

Problem 1

If you get stuck, consider using coins or other circular objects.

  1. How many circles of radius 1 unit can you fit inside a circle of radius 2 units so that they do not overlap?

  2. How many circles of radius 1 unit can you fit inside a circle of radius 3 units so that they do not overlap?

  3. How many circles of radius 1 unit can you fit inside a circle of radius 4 units so that they do not overlap?

Activity 2: Covering a Circle

Problem 1

Here is a square whose side length is the same as the radius of the circle.

A circle with radius shown and a square with the same side length as the radius.

How many of the squares do you think it would take to cover the circle exactly?

Lesson Summary

The circumference of a circle is proportional to the diameter , and we can write this relationship as . The circumference is also proportional to the radius of the circle, and the constant of proportionality is because the diameter is twice as long as the radius. However, the area of a circle is not proportional to the diameter (or the radius).

The area of a circle with radius is a little more than 3 times the area of a square with side so the area of a circle of radius is approximately . We saw earlier that the circumference of a circle of radius is . If we write for the circumference of a circle, this proportional relationship can be written .

The area of a circle with radius is approximately . Unlike the circumference, the area is not proportional to the radius because cannot be written in the form for a number . We will investigate and refine the relationship between the area and the radius of a circle in future lessons.