Investigate the relationship between surface area and volume of cuboids

Question

• How can you change the surface area of a cuboid while keeping the volume the same?

• How can you change the volume of a cuboid while keeping the surface area the same?

Make conclusions by generalising – what can you say about the relationship between surface area and volume of cuboids in general?

Answer 1

A cube is the cuboid that minimizes the surface area compared to the volume. So, changing side lengths from a cube (maintaining angles at 90: rectangular cuboid) while maintaining volume will increase surface area:

4x4x4 = 64 for volume and 6(4²) = 96 for surface area

8x2x4 is the same volume, but the surface area = 2(8x2+2x4+4x8) = 112

Of course going from a rectangular cuboid to a cube (or closer to a cube), you can maintain surface area and maximize volume. In the example given 112 SA could go to a cube with side (112/6)^1/2 = 4.32 which has a volume of 80.6.

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