# Lesson 7

Exploring the Area of a Circle

Let’s investigate the areas of circles.

### 7.1: Estimating Areas

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

### 7.2: Estimating Areas of Circles

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 $$ABC$$, click on $$A$$-$$B$$-$$C$$-$$A$$. 2. Record the diameter in column $$D$$ and the corresponding area in column $$A$$ for your circles and others from your classmates.
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?

How many circles of radius 1 unit can you fit inside each of the following so that they do not overlap?

1. a circle of radius 2 units?
2. a circle of radius 3 units?
3. a circle of radius 4 units?

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

### 7.3: Covering a Circle

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

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

### Summary

The circumference $$C$$ of a circle is proportional to the diameter $$d$$, and we can write this relationship as $$C = \pi d$$. The circumference is also proportional to the radius of the circle, and the constant of proportionality is $$2 \boldcdot \pi$$ 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 $$r$$ is a little more than 3 times the area of a square with side $$r$$ so the area of a circle of radius $$r$$ is approximately $$3r^2$$. We saw earlier that the circumference of a circle of radius $$r$$ is $$2\pi r$$. If we write $$C$$ for the circumference of a circle, this proportional relationship can be written $$C = 2\pi r$$.

The area $$A$$ of a circle with radius $$r$$ is approximately $$3r^2$$.  Unlike the circumference, the area is not proportional to the radius because $$3r^2$$ cannot be written in the form $$kr$$ for a number $$k$$. We will investigate and refine the relationship between the area and the radius of a circle in future lessons.

### Glossary Entries

• area of a circle

If the radius of a circle is $$r$$ units, then the area of the circle is $$\pi r^2$$ square units.

For example, a circle has radius 3 inches. Its area is $$\pi 3^2$$ square inches, or $$9\pi$$ square inches, which is approximately 28.3 square inches.