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
Your teacher will give your group two circles of different sizes.
- Set the diameter of your assigned circle and use the applet to help estimate the area of the circle. \(ABC\), click on \(A\)-\(B\)-\(C\)-\(A\). 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
- Record the diameter in column \(D\) and the corresponding area in column \(A\) for your circles and others from your classmates.
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?
- a circle of radius 2 units?
- a circle of radius 3 units?
- 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?
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.
- 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.