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 \(A=πr^2\), 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.
Lesson overview
 7.1 Warmup: Estimating Areas (5 minutes)

7.2 Activity: Estimating Areas of Circles (20 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 7.3 Optional Activity: Covering a Circle (20 minutes)
 Lesson Synthesis
 7.4 Cooldown: Areas of Two Circles (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s investigate the areas of circles.
Learning targets (student facing):
 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.
Required materials:
 copies of blackline master
 geometry toolkits
Required preparation:
 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).
Glossary:
 area of a circle  If the radius of a circle is \(r\) units, then the area of the circle is \(πr^2\) 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.
 Access the complete Grade 7 glossary.
Standards:
 This lesson builds towards the standard: CCSS.7.G.B.4MS.7.G.4MO.7.GM.A.4b
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses.