This lesson is the first of two lessons that develop the formula for the area of a circle. Students start by estimating the area inside different circles, deepening their understanding of the concept of area as the number of unit squares that cover a region, and discovering that area (unlike circumference) is not proportional to diameter.
Next, they investigate how the area of a circle compares to the area of a square that has side lengths equal to the circle’s radius. Students may choose tools strategically from their geometry toolkits to help them make these comparisons (MP5). Students find an approximate formula: the area of a circle is a little bigger than \(3r^2\), and they check their earlier estimates with this formula. At this point, it is a reasonable guess that the exact formula is \(A=\pi r^2\), but the next lesson will focus on using informal dissection arguments to establish this formula.
When we say “area of a circle” we technically mean “area of the region enclosed by a circle.” However, “area of a circle” is the phrase most commonly used.
- Create a table and a graph that represent the relationship between the diameter and area of circles of various sizes, and justify (using words and other representations) that this relationship is not proportional.
- Estimate the area of a circle on a grid by decomposing and approximating it with polygons.
Let’s investigate the areas of circles.
For the first activity, you will need the Estimating Areas of Circles blackline master. Prepare 1 copy for every 12 students. (Each group of 2 students gets one of the pages.)
For the second activity, make sure students have access to their geometry toolkits, especially tracing paper and scissors, if they so choose (but try not to influence students' choices about what tools to use).
- 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.
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.