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I was considering just using the name rectangle for representing the set of points contained in a 3D plane for a given rectangular area.

I would like to know whether there's a more appropriate name. If I understood correctly, by definition, that would not be a finite plane.

UPDATE I know the difference between disk and circle: the disk's set of points includes points inside the circular region.

Is there something equivalent for rectangles?

LRMAAX
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    In my experience, a circle is the set of points on the circumference. The set that also includes points in the interior is called a disk. –  Oct 16 '18 at 17:20

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It's fine to use rectangle; the context should make it clear whether you're talking about the rectangular region itself or just the perimeter (and this sentence shows how to distinguish the two if necessary).

Théophile
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You might want to take refuge to attributes here. Then you could distinguish between an "open rectangular region" and a "closed rectangular region". And sure there is the "rim of that rectangular region".

Wrt. some finite 2D fabric you also could take refuge alongside to the greekish term "polyhedron", i.e. some solid having many (poly) seets (hedra). Accordingly a finite 2D fabric might well be termed some "hedrix". - Cf. here also the Polygloss of Wendy Krieger, http://os2fan2.com/gloss/pglossh.html, which was piled up right for the purpose to dig out the ethymological stems of terms and to base dimensional extrapolations of those in a ethymological correct sense.

--- rk

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The thesaurus suggested slab, pane, and lamina; among others.