I may be wrong, but I think you're expecting the Area tool to place the measurement info into something within the drawing. All it does is generate the info into the Measurement Info palette. From there you can copy-n-paste it into the Leader tool or a Text tool.
Henry's way is nice, but you don't have easy access to the info if you wish to copy-n-paste like you can with the Measurement Info palette. Some people aren't up to speed with scientific notation either.
If you do want to use the Area tool, choose the "By Entity" parameter (Local Menu), select the polyline, then select "Close". The info should appear in the palette. You could also do this without "By Entity", but you would snap to the points along the polyline. If you have overlapping polylines (as in the lower section), you could use "By Entity", then snap to the points. This takes a bit getting used to. The little blurb in Help about Measuring Area
may be useful if you haven't seen it yet. Attached is your drawing with leaders noting the square footage copied out of the palette. I used " \P
" to force a line break in the Leader text.
Designer, Deluxe, (Professional, Expert, Basic), Platinum
Windows 10 Pro (1803), 64-bit