Select the path
Cmd-Shift-Option-F12 to open the secret programmer's debugging palette
In that palette, scroll down until you see a section called "object tree"
Click on the 'path' label that is in bold-face (which indicates the object is selected)
The "objects" section above will then display various data about the path, including its area in square pts. Negative areas indicate counter-clockwise paths, positive areas are clockwise.
If the object you wish to measure is not near the top of its layer in the stacking order, you may wish to duplicate it to the top via Copy, deselect, Paste in Front, since it can be hard to find lower-down objects in the debug palette window.
Warning: This palette is not supported. It intentionally does not process many notifications, and can therefore cause crashes if you leave it open while editing the document, especially edits that delete objects which it is displaying data about.
...further calculation is required, i think it is this, though don't quote me on it as i'm crap at maths:
(value in pts / 72 squared) = value in inches) x 25.4 = value in millimetres
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