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A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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Sending a Parcel

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?


Age 11 to 14 Challenge Level:

Why do this problem?

This problem requires a lot of calculations of surface areas, within a rich problem solving context.

Possible approach

This printable worksheet may be useful: Cuboids.

Work with a specific cuboid, eg $2 \times 3 \times 5$, or a breakfast cereal box, to establish how to calculate surface area of cuboids. Students could practise working out surface area mentally on some small cuboids made of multilink cubes.

Present the problem, ask students to keep a record of things that they tried that didn't work (and what was wrong) as well as things that did work. In this initial working session, try to ensure that students are calculating surface area correctly. This spreadsheet may be useful (for teachers' eyes only!).

It may be appropriate to draw a ladder on the board, with this on the steps (starting from the bottom):
- calculations going wrong
- no solutions yet
- one solution
- some solutions
- all solutions
- why I am sure I have all the solutions
- I'll change the question to...

Explain to students that wherever they are on the ladder, their goal is to move up to the next step. Circulate round the class and look out for students who are approaching the task systematically.

After students have been working for a little while, bring the class together and ask them for strategies that might help others move up the ladder.
Give them plenty of time to implement these suggestions. 

This might be a good lesson in which to allocate five minutes at the end to ask students to reflect on what they have achieved, which methods and ideas were most useful, and what aspects of the problem remain unanswered.

Key questions

  • Have you found none/one/some or all of the solutions
  • Is there a cube that will work?
  • How might you organise a systematic search for the cuboids with surface area $100$?

Possible extension

The main extension activity could focus on the convincing argument that all solutions have been found. Once this has been answered, you might like to consider these extensions:
  • Express the method for calculating surface area, algebraically.
  • Which surface area values will generate lots of cuboids, and which give none or just one?
  • Could you set up a spreadsheet to help with the calculations?

Possible support

Suggest students approach the task systematically:
if the height is $1$, what are the possible combinations for the width and depth?
if the height is $2, 3, 4$... what are the possible combinations for the width and depth?

In groups, or as a class, keep a record of all cuboids whose surface areas have been calculated.
Award ten points for a bulls eye "$100$", five points for each $95-105$, and two points for $90-110$.
Any miscalculated results could lose points, providing motivation for peer checking, and helping each other.

A sheet showing a net of a cuboid, like this , may help students to organise their working and ideas.