Illuminations: Circle Packing

# Circle Packing

In this unit, students explore circles. In the first lesson students apply the concepts of area and circumference to explore arrangements for soda cans that lead to a more efficient package. They then experiment with three-dimensional arrangements to discover the effect of gravity on the arrangement of soda cans in the second lesson. The final lesson allows students to examine the more advanced mathematical concept of curvature.

### Math Content

In this unit, students will:
• Determine circumference and area.
• Calculate the efficiency of an arrangements of cans
• Create a model for arranging cans on a shelf.
• Determine the curvature of a circle.

### Prerequisite Knowledge

Students should be able to calculate circumference and area.

### Individual Lessons

Lesson 1 - Soda Cans

Soda cans are often packaged in rectangular arrays, but more efficient arrangements that require less packaging material are possible. In this lesson, students investigate various designs for packaging soda cans and use geometry to analyze their designs.

Lesson 2 - Soda Rack

In the previous lesson, students considered an arrangement of cans in which the cans were placed on a shelf. In this lesson, students consider the arrangement of cans placed in a bin with two vertical sides, discover an interesting result, and prove that the result is true.

Lesson 3 - Circle Packing and Curvature

An important idea in advanced mathematics is curvature, the amount by which a geometric object deviates from being flat. Mathematicians study the curvature of advanced curves and three-dimensional shapes. In this lesson, students investigate the curvature of circles.

### NCTM Resources

 More and Better Mathematics for All Students
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