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A line extends to infinity, a line segment is a portion of a line. A plane is infinite. What is a portion of a plane called ? A plane segment?

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There's only one possibility in the case of a line segment, but not in the case of a plane... so we call them squares, rectangles, triangles, etc., etc., etc...  you could call it a "plane shape" if you want...
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Assuming we’re talking about “rectangular” subsets of the (x,y) plane, here’s what I’ve got:

Neither x nor y bounded: plane

Only one of x or y bounded in only one direction: half-plane

Both x and y bounded in only one direction: quarter-plane

Either x or y bounded in both directions, the other unbounded: strip

Either x or y bounded in both directions, the other bounded in one direction: half-strip

Both x and y bounded in both directions: rectangle or box

Please note that I’m just making these up, but they seem reasonable to me.
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Any closed region
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A region
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There are a number of potential answers to a question like this.

Most generally, you could refer to a region of a plane. A region could really be any sort of shape and could be unbounded, you you might impose various conditions on it (e.g. boundedness, compactness, connectedness,  convexity, etc).

But even if you want to stick to analogs of line segments, rays, and lines, there are some options. In a certain sense, you can think of planes as a kind of product of two (non-parallel) lines. In the same sense, a half plane would be the product a line and a ray (again, non-parallel, which I'll stop specifying from here). We can consider regions which are products of each combination of line segments (S), rays (R), and lines (L).

* S×S → A parallelogram.  
* S×R → Like a parallelogram, but two opposite sides extend indefinitely in one direction, as rays.  
* S×L → A region enclosed by two parallel lines.  
* R×R → A quarter plane.  
* R×L → A half plane.  
* L×L → A plane.

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