Cuboids and the smallest surface area

This picture shows a rectangular prism made from 36 cubes. How many different rectangular prisms (cuboids) can you make from 36 cubes? Do they all have the same volume? Why or why not?

Look at the date today, for example suppose it is 11th of the month. How many different cuboids can you make using 11 small cubes?

The answer is that you can only make one, a long rod measuring 1 by 1 by 11 because 11 is a prime number, but there are some cuboids made from 12 cubes with one left over? How many? Whatever the date, can you answer the same questions?

Think about a cuboid like the one pictured above and all the cuboids made from the same number of cubes. Do they all have the same surface area? Why or why not?

Take the area of each face of the cube as one square unit.
Find the surface area of all the rectangular prisms that can be made using 36 small cubes.
You will have to work systematically and take note of your results.
Which one has the smallest surface area?

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