What are the dimensions of area?
We know that, in general, an area can have many different shapes. An area could be a square or a triangle or a circle or some other shape. And yet our claim is that, in all these cases, the dimensions of area are the same. If we call our area 𝐴, the dimensions of 𝐴 are not units, but rather the quantities that together express an area.
We can start to understand what these dimensions are if we imagine that our square has sides of length 𝑠, our triangle has a height ℎ and a base 𝑏, and our circle has a radius 𝑟. Knowing these quantities, we can say that the area of our square is 𝑠 squared, the area of the triangle is one-half its base times its height, and the area of the circle is 𝜋 times 𝑟 squared. Notice that in all three of these cases, we have a length being multiplied by another length. This is an essential quality of an area.
Ignoring any constant factors, an area is basically a length multiplied by another length. We could also write this as 𝐿 squared. The dimensions of area then are length squared.