this question has been bugging me for a while now! I was drawing boxes over some squares in a book and then wrote 1cm and 10mm for the line length, and when I calculated the area I realised that the areas were not the same but they occupied the same amount of space.
Am I missing something obvious or is there an explanation?
Attached is a diagram of what I mean. All squares of the same dimensions for mm, cm and m.
On
If length is measured in millimeters ($\textrm{mm}$), then area is measured in square millimeters ($\textrm{mm}^2$).
If length is measured in centimeters ($\textrm{cm}$), then area is measured in square centimeters ($\textrm{cm}^2$).
One square centimeter ($1 \textrm{ cm}^2$) is 100 square millimeters ($100 \textrm{ mm}^2$).
And similarly where the unit is meters.
I think you are confused because you have used linear units instead of areal units when you wrote down the area. You computed it correctly, just didn't write the correct units for the numerical answers.
You haven't made a mistake, and the areas are equal. It's just that $100mm^2 = 1cm^2 = 0.0001m^2$ is the actual relation between the different area units. What you have done here is a correct proof of that fact.