The typical way that area is communicated is of the form “area units^2”, for example:
“California fire officials say a wildfire in Northern California has destroyed more than 6,500 structures and grown to 140 square miles (362 square kilometers).”
source: News Channel 9, https://newschannel9.com/news/nation-world/north-california-fire-kills-9-destroys-6700-structures-grows-to-140-square-miles
“In the US, surface car parks take up land roughly about the size of Puerto Rico (which is 3,459 sq miles or 8,959 sq km), according to conservative estimates in a new book by Eran Ben-Joseph, professor of landscape architecture and planning at Massachusetts Institute of Technology (MIT). He thinks the true figure could be up to 4,700 sq miles (12,170 sq km, larger than Jamaica or Qatar).”
source: BBC, https://bbc.in/2PScHPp
I don’t find area very intuitive in the same way that distance is. When I read articles like those above, in order to actually visualize and get a ‘feel’ for the sizes involved, I often try to turn the area into a square/rectangle in my mind:
140 square miles: my first reaction is to visualize a 14 x 10 mile rectangle. I can then relate this to where I live and then understand what size is involved.
4,700 square miles: using similar logic, I think of 47 x 100. This is less like a square so I have to think a bit more, but at least I can relate to it. Finally, if I’m really wanting to get to the bottom of the area involved, I will take the square root – in this case the area relates to a square 68.55 x 68.55 miles. I can immediately visualize this in relation to where I live or other parts of the world I know where there is roughly 70 miles between places.
Therefore this leads me to beg the question: would the format “x^2 units^2”, e.g “69^2 square miles” or “sixty-nine squared, square miles” not be a better format for reporting area in many situations?
Granted, there will always be a use for reporting area as “x units^2” as sometimes it is useful to think of area in that way, e.g. “100kg of fertilizer are required per square km, and the fields cover 10km therefore 1,000kg of fertilizer are required.” However, equally, surely there could be a use for reporting “x^2 units^2” especially in situations where computations aren’t required and the purpose is really to convey the magnitude of an area?