# Juice Box Geometry

A fun, fruity math challenge from Science Buddies

Biggest Box?: Some simple geometry can help you figure out which box will hold the most liquid--and which has the most efficient packaging. Image: George Retseck

Key concepts
Mathematics
Geometry
Psychology

Introduction
Juice boxes sure are convenient—just poke a straw in and sip away! But have you ever noticed that some juice boxes don't seem to have as much juice as others, even when they have a lot of packaging?

You might be amazed how much thought goes into designing a juice box. Each manufacturer has carefully calculated how big each side of the box needs to be to hold a certain amount of juice inside and how altering the box's dimensions affects its overall appearance and packaging.

Investigating the geometry of juice boxes can reveal how some have more efficient packaging than others. And how appearances can be deceptive!

Background
Geometry uses math to describe and investigate different points, lines and shapes. In geometry, the relationships among different parts of a shape can be described with mathematical formulas. Formulas can be used to describe the relationship among the three dimensions of real-world objects: length, width and height. And that information can help us learn about objects around us.

A rectangular box, such as a juice box or cereal box, is called a rectangular prism in geometry-speak. 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 that object. The volume is the amount of space an object fills in three-dimensional space (which tells approximately how much a container, like a juice box, can hold), and the surface area is the total amount of area on the outer surface of the object (which tells approximately how much material was used to create the shape).

Materials
•    Metric ruler
•    Juice boxes, five or more different brands (one box per brand)
•    Calculator
•    Paper
•    Pen or pencil

Preparation
•    Using a ruler, for each juice box measure its length (how long it is from left to right), width (how deep it is from front to back) and height. Write down these measurements for each box (in centimeters). Are the dimensions fairly similar between the different boxes, or is there a lot of variation?

Procedure
•    For each juice box, calculate the surface area and write it down. The surface area of a rectangular prism = 2 X length X height + 2 X width X height + 2 X length X width. (The units will be in cm2, or centimeters squared.) How much do the surface areas vary for the different boxes?
•    For each juice box, calculate the volume and write it down. The volume of a rectangular prism = length X height X width. (The units will be in cm3, or centimeters cubed) How much do the calculated volumes vary for the different boxes?
•    Compare the volume of juice in each box with its calculated volume. The volume of juice in the box is usually found on the front of the juice box near the bottom. Use the volume listed in milliliters (mL) because 1 mL equals 1 cm3. How close is your calculated volume to the actual volume of juice in each box?
•    For each juice box, calculate the ratio of surface area to volume. You can do this by dividing its calculated surface area by the volume of juice it holds (again in milliliters). If the answer of the calculation is close to 1, 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. Which box uses the least amount of packaging for the volume of juice it holds? Which box uses the most?
•    Overall, which juice box is the most efficient at holding juice for its size? Which one has a juice volume closest to its calculated volume and has the smallest surface-area-to-volume ratio?
•    Extra: Packaging comes in all shapes and forms. Try to use the principles of geometry to make geometric models to compare the efficiency of other shapes of packaging. For example, how do cylindrical ice cream tubs compare with rectangular ice cream tubs?
•    Extra: Cereal containers mostly come in rectangular shapes but can vary greatly in size. You can use different cereal boxes to investigate how geometry can help reduce waste in packaging material. Is there less wasted packaging in a large box or a small box of cereal? Is it better to buy single-use cereal boxes that are only one portion size or large, multi-portion size boxes?

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