Lesson 7

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.

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).