Why is perimeter measured in units and area measured in square units?

Answer 1

When you're measuring perimeter you are only dealing with one dimension. If I asked you to find the perimeter of a square whose side measures 2 m, you would instinctively say it was 8 m. You would probably show your work like this:
p=4(2)
p=8 m

Now do it, but carry the units m throughout the calculation.
p=4(2 m)
p=8 m

In one dimension, this is trivial. Let's try in two. Find the square's area. Once again carry the units.
A=(2 m)(2 m)
A=4 m²

When multiplying meter by meter (m times m), you get m². The same principle applies to volume.
v=(2 m)(2 m)(2 m)
v=8 m^3