Uniform units

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Age
14 to 16
Challenge level
filled star filled star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



 


I have a $1$m cube. Its volume is $1$m$^3$, its surface area $6$m$^2$ and its total edge length $12$m. If I measure the dimensions of the same cube in centimetres, then its volume would be $1000000$cm$^3$, its total surface area $60000$cm$^2$ and its total edge length $1200$cm.

In the first case, the total edge length is numerically largest, whereas in the second case the total volume is numerically largest.

Could you choose a unit of measurement so that the surface area was numerically largest?

Can you choose a unit of measurement so that two of the quantities are numerically the same?

Why can't you find a unit of measurement so that all three quantities are numerically the same?

Can you invent any shape for which you can choose units so that the total edge length, total surface area and volume are numerically equal?