The Mathematics of a Soda Pop Can

This HiMAP Pull-Out Section explores the mathematics behind how a company chooses the size and shape of its product container. Is the can the most efficient shape for the product? Is the container the cheapest way the company can make its product? Tum the page and find out.

Note: The information below was created with the assistance of AI.

Level of Mathematics

This module is best suited for:

• High school students in grades 10–12, particularly those in Algebra II, Geometry, or Precalculus.

• It is especially effective as an enrichment activity in honors-level math or in STEM-focused curricula.

• Optional problems require calculus, making it adaptable for AP Calculus AB students.

It combines algebraic reasoning, geometry, and optimization concepts, bridging visual thinking with analytical problem-solving.

Application Areas

This lesson connects mathematics to multiple real-world and interdisciplinary domains:

• Product design & manufacturing: Investigating efficient shapes and material use for containers.

• Industrial engineering: Surface area minimization and material cost analysis.

• Marketing and design: Exploration of aesthetics through the “golden mean”.

• Consumer product analysis: Comparing surface area and volume of commercial packaging.

• Environmental science: Reducing material use (aluminum) in packaging can reduce waste and cost.

It uses a common, relatable object (a soda can) to drive engagement and application.

Prerequisites

To fully benefit from this material, students should have experience with:

• Algebraic manipulation

• Geometry: Area and volume formulas for cylinders and rectangular solids.

• Square roots and rational exponents

• Using inequalities in proofs

• Familiarity with optimization (for the non-calculus parts)

• Conceptual understanding of ratios, including the golden mean

Optional parts assume familiarity with derivatives and critical points.

Subject Matter

1. Geometry of Containers

• Volume and Surface Area:

  • Cylinder: V=πr2hV = \pi r^2 hV=πr2h, S=2πr2+2πrhS = 2\pi r^2 + 2\pi r hS=2πr2+2πrh

  • Rectangular box: $V=xyz$, $S=2(xy+yz+xz)$

• Golden Mean: Aesthetic proportions, defined algebraically as y+xy=yx≈1.618\frac{y + x}{y} = \frac{y}{x} \approx 1.618yy+x​=xy​≈1.618

2. Optimization

• Minimizing surface area for a fixed volume.

• Use of Arithmetic Mean ≥ Geometric Mean (AM-GM Inequality):

  • For 2 variables: x+y2≥xy\frac{x + y}{2} \geq \sqrt{xy}2x+y​≥xy​

  • For 3 variables: x+y+z3≥xyz3\frac{x + y + z}{3} \geq \sqrt[3]{xyz}3x+y+z​≥3xyz​

3. Algebraic Proofs and Inequalities

• Development and application of AM-GM to practical design problems.

• Application to real-world packaging efficiency.

4. Comparison of Designs

• Comparison of cylindrical vs. cube-shaped containers in terms of surface area.

• Exploration of why cans are not made using the mathematically optimal dimensions (e.g., human ergonomics, marketing).

Correlation to Mathematics Standards
Common Core High School Standards

Geometry

• HSG-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects.

• HSG-GMD.A.1: Give an informal argument for formulas for the volume of cylinders and rectangular solids.

• HSG-GMD.A.3: Use volume formulas to solve real-world and mathematical problems.

Algebra

• HSA-SSE.A.1–2: Interpret expressions and their structure.

• HSA-REI.B.3: Solve equations and inequalities algebraically.

Mathematical Practices

• MP1: Make sense of problems and persevere in solving them.

• MP2: Reason abstractly and quantitatively.

• MP4: Model with mathematics.

• MP6: Attend to precision.

• MP7: Look for and make use of structure.

(Optional) Calculus Standards

• Rate of change and optimization via First/Second Derivative Test for surface area minimization.