# Juice Box Geometry

## Abstract

Juice boxes are so convenient—just poke the straw in and sip away! But have you ever noticed that some juice boxes don't seem to have much juice, even when they have a lot of packaging? It might surprise you how much thought goes into the design and manufacturing of a juice box. Each manufacturer has carefully calculated how big each side should be to hold a certain amount of juice inside. In this science project, you will find out how different brands of juice measure up.## Summary

Sara Agee, Ph.D., Science Buddies

Sandra Slutz, Ph.D., Science Buddies

## Objective

Measure the dimensions (size in different directions such as height, length, and width) of different juice box products to find out which manufacturer has the largest volume of juice and uses the least amount of packaging material.

## Introduction

**Geometry** is the study of how to use math to describe and investigate different points, lines, and shapes. The way that a shape is described in geometry is with a mathematical calculation called a **formula**. Formulas can be used to describe the relationships among different parts of a shape, including its **length**, **width**, and **height**.

A rectangle drawn on a piece of paper is a **two-dimensional shape**. This means it has two dimensions: height and length. But real-world objects are **three-dimensional shapes** (sometimes called *solids*). They have three dimensions: height, length, and width. A rectangular box, like a juice box or cereal box, is called a **rectangular prism** in geometry.

If you measure the height, length, and width of a rectangular prism, you can use that information in a formula to calculate the volume and surface area of the container. The **volume** is the amount of space an object fills in three-dimensional space. It also tells you approximately how much a container, like a juice box, can hold. The **surface area** is the total amount of area on the outer surface of the object. It also tells you approximately how much material was used to create the shape.

In this experiment, you will use geometry to produce a mathematical model of a juice box. You will measure rectangular prisms (juice boxes), and use formulas to discover approximately how much juice each box can hold (volume) and how much packaging (surface area) was used. Which brand do you think holds the most juice? Which uses the least amount of packaging?

### Terms and Concepts

To do this type of experiment you should know what the following terms mean. Have an adult help you search the Internet, or take you to your local library to find out more!

- Geometry
- Formula
- Length (l)
- Width (w)
- Height (h)
- Two-dimensional shape
- Three-dimensional shape
- Rectangular prism
- Volume (V)
- Surface area (S)

### Questions

- If you had two juice boxes, both with the same volume, but one had more packaging than the other, which would be better? Why?
- What other three-dimensional shapes do you see in food packaging?

### Bibliography

This science project uses this online program to calculate the surface area and volume of a rectangular prism:

- Calculator Soup. (2010). Rectangular Prism Calculator. Retrieved June 13, 2011.

This site gives you fun ways to review math skills, solve puzzles, and read stories about math. The site also has a section on geometry:

- Coolmath.com Inc. (2009) Cool Math 4 Kids. Retrieved June 13, 2011.

For help creating graphs, try this website:

- National Center for Education Statistics. (n.d.). Create a Graph. Retrieved June 13, 2011.

## Materials and Equipment

- Metric ruler
- Juice boxes, five or more different brands (one per brand)
- Computer with Internet connection
- Lab notebook

## Experimental Procedure

- In this experiment, you will be measuring the sides of several juice boxes to calculate the amount of packaging that each juice box is made of (surface area) and how much juice it holds (volume). Make a data table, like Table 1 below, in your lab notebook to keep track of your measurements and other data.

Brand Name | Volume in Ounces (oz) | Length (cm) | Width (cm) | Height (cm) | Surface Area (cm^{2}) |
Calculated Volume (cm^{3}) |

**Table 1.**You will need a data table, like this one, to keep track of all your juice box measurements.

- Take the first juice box and write in your data table:
- The brand name of the juice box.
- The volume of juice (in fluid ounces) that the manufacturer put in the box. This information is usually found on the front of the juice box near the bottom.

- Using a metric ruler, measure the length, width, and height of the juice box in centimeters, and write the measurements into the data table. Figure 1 in the Introduction shows you where to make each of these measurements.
- Using your computer, click the online Calculator Soup program to calculate the volume and surface area of a rectangular prism as shown below.
*Note:*The full citation is listed in the Bibliography. - Type in your values for length, width, and height into the boxes, choose your units (this should be centimeters), and click the
*Calculate*button.- When you click the Calculate button, the online program uses the length, height, and width measurements in the formulas for the surface area of a rectangular prism (see Equation 1) and the volume of a rectangular prism (see Equation 2).
- If you know how to multiply, you can try using the formulas in Equations 1 and 2 yourself and see if you agree with the online calculator's answers.

**Equation 1.**

**Surface area of a rectangular prism** = 2 × length × height + 2 × width × height + 2 × length × width

**S**is the surface area in square centimeters (cm^{2})**l**is the length of the prism in centimeters (cm)**h**is the height of the prism in centimeters (cm)**w**is the width of the prism in centimeters (cm)

**Equation 2.**

**Volume of a rectangular prism** = length × height × width

**V**is the volume in cubic centimeters (cm^{3})**l**is the length of the prism in centimeters (cm)**h**is the height of the prism in centimeters (cm)**w**is the width of the prism in centimeters (cm)

- Write the data for volume (V) and surface area (S) into your data table. The volume will be in cubic units (cm
^{3}), and the surface area will be in squared units (cm^{2}). - Repeat steps 2–7 for each of the four remaining juice boxes.

### Analyzing and Graphing the Data

- Make a bar graph of surface area and volume for each brand of juice box. You can make your graphs by hand, or you can try using the Create a Graph web site for kids from the National Center for Education Statistics.
- Which brands use the least amount of packaging material? Compare the fluid ounces to calculated volume. How do they compare? Are they the same? Which brands give you the most juice per juice box?

## Ask an Expert

## Variations

- This science project works for containers that are 3-D rectangular prisms, but packaging comes in all shapes and forms. Can you use the principles of geometry to investigate other forms of packaging? Can you make geometrical models for other products? How do cylindrical ice cream tubs compare to rectangular ice cream tubs?
- A more advanced way to compare the different brands is to calculate the ratio of Surface Area to Volume. Sometimes the ratio is written as S:V, and sometimes it is expressed as a fraction, S/V. Either way, compute this calculation by dividing the surface area by the volume. If your answer is close to one, the surface area and volume are almost equal. The higher the number, the more packaging is used per volume of juice. The lower the number, the less packaging is used per volume of juice. When you compare the different ratios, which brands
*minimized*the S:V ratio to reduce packaging costs? - In this science project, you are using containers that are basically the same size and that hold juice. What about other packages for other products? How can geometry help reduce waste in packaging material? Is there less wasted packaging in a large box or a small box of cereal? How about single-use cereal boxes that are only one portion size? Is it better to buy individually wrapped raisin boxes or a big box of raisins?

## Careers

If you like this project, you might enjoy exploring these related careers:

## Related Links

## Cite This Page

General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.#### MLA Style

*Science Buddies*, 21 Oct. 2021, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p020/pure-mathematics/juice-box-geometry?from=Home. Accessed 26 Nov. 2022.

#### APA Style

*Juice Box Geometry.*Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p020/pure-mathematics/juice-box-geometry?from=Home