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Learning Targets
- If I know a circle’s radius or diameter, I can find an approximation for its area.
- I know whether or not the relationship between the diameter and area of a circle is proportional and can explain how I know.
Notes
Notes
Let’s investigate the areas of circles.
The circumference C of a circle is proportional to the diameter d, and we can write this relationship as C = π d. The circumference is also proportional to the radius of the circle, and the constant of proportionality is 2 ⋅ π 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 3r2. We saw earlier that the circumference of a circle of radius r is 2 π r. If we write C for the circumference of a circle, this proportional relationship can be written C = 2 π r.
The area A of a circle with radius r is approximately 3r2. Unlike the circumference, the area is not proportional to the radius because 3r2 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.
Vocabulary
Vocabulary
area of a circle: The area of a circle whose radius is r units is π r2 square units.
Activities
Activities
7.2 Estimating Areas of Circles
7.2 Estimating Areas of Circles
Open both applets ->
- Pick a pair of diameters from the list below.
- Set the diameter of your 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.
- 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?
- 2 cm and 16 cm
- 5 cm and 10 cm
- 3 cm and 12 cm
- 6 cm and 18 cm
- 4 cm and 20 cm
- 7 cm and 14 cm
Things to Note
Things to Note
There is a proportional relationship between diameter and circumference, even though there is not between diameter and area. Recall that you saw the same phenomenon when you examined the relationship between the diagonal of a square and its perimeter (proportional) and the diagonal of a square and its area (not proportional).
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