What it shows
The same 24 sugar cubes or dice can be stacked into several boxes with the same volume, but their outside areas differ a lot. The box closest to a cube, 2 by 3 by 4, has the smallest surface area, which is why packing and animal shapes matter.
The science
Volume of a cuboid is length times breadth times height, and surface area is 2(lb + bh + lh). For a fixed volume, the surface area is smallest when the box is as close to a cube as possible. A long thin box has more outside faces exposed. This idea explains why a round pot keeps a curry warmer than a flat tray, and why small animals lose heat faster than large ones.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- 24 identical small cubes, such as dice, wooden blocks or sugar-cube-sized clay blocks
- Squared paper with 1 cm squares
- A pencil and ruler
- A notebook
- A calculator
Safety
For this project: Use clean blocks and keep small blocks away from young children who may put them in their mouths.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
Step by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
List all the ways to make a box with 24 cubes: 1 by 1 by 24, 1 by 2 by 12, 1 by 3 by 8, 1 by 4 by 6, 2 by 2 by 6 and 2 by 3 by 4.
Build each box with the cubes and put the list on a table. Check the number of cubes is still 24 each time.
For each box, count the number of square faces on the outside by turning it over, or calculate with 2(lb + bh + lh).
Write the results in a table with the dimensions, volume and surface area. For example 1 by 1 by 24 has 98 square units.
Check your numbers on one box by drawing its net on squared paper and counting all the squares.
Find which shape has the smallest area, which should be 2 by 3 by 4 with 52 square units, and the largest, 98.
Draw a bar chart of the surface area for the six boxes.
Write a short note on why a cuboid with a smaller surface is better for packing a given amount of goods, and why lumps of sugar are cubes.
What you should see
The six boxes all hold 24 cubes but have surface areas of 98, 76, 70, 68, 56 and 52, with the 2 by 3 by 4 box smallest.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What is the formula for the surface area of a cuboid?
2 times (length times breadth + breadth times height + length times height).
02Does the volume change from box to box?
No, all have 24 cubic units.
03Which shape has the least surface for a given volume?
A sphere, and among cuboids the cube or nearly cube shape.
04Why is this useful to a packer?
A compact shape needs less packing material.
Take it further
Repeat with 36 cubes and list all boxes, then predict the best before calculating.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
Free ebooks for this project
Science projects
Maths Working Models and Activities
Make maths visible: models for number, geometry, algebra and data
Science projects
Science Projects for Class 6 to 8
Working models matched to middle-school science topics
Find out more about maths
Free, well-known places to explore the subject further. Ask a grown-up before you sign up for anything or type your name anywhere.