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'Changing Areas, Changing Volumes' printed from http://nrich.maths.org/
This problem follows on
from Changing Areas, Changing
Perimeters.
Here are the dimensions of nine cuboids. You can download a set of
cards
here.
1 by 2 by 28
cuboid

4 by 4 by 4
cube

2 by 4 by 7
cuboid

1 by 2 by 26
cuboid

2 by 4 by 6
cuboid

4 by 5 by 6
cuboid

4 by 5 by 7
cuboid

1 by 2 by 24
cuboid

1 by 4 by 14
cuboid

The challenge is to arrange them in a 3 by 3 grid like the one
below:
As you go from left to right, the surface area of the shapes must
increase.
As you go from top to bottom, the volume of the shapes must
increase.
All the cuboids in the middle column must have the same surface
area.
All the cuboids on the middle row must have the same volume.
What reasoning can you use to help
you to decide where each cuboid must go?
Once you've placed the nine cards, take a look at the extended grid
below:
The ticks represent the nine cards you've already placed. Can you
create cards with dimensions for cuboids that could go in the four
blank spaces that satisfy the same criteria?
Can you design a set of cards of your own with a different cuboid
in the centre?