Anything that has two dimensions and is completely closed is a shape (as opposed to a line which as one dimension, or a point which has zero). A property of these shapes will be that they all have length and height with which area can be calculated.
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Any shape is mathematical.
Any shape tesselates any of its covering spaces.
If you mean a quadrilateral shape then it is any shape that has 4 sides.
Any closed shape in 2-dimensions can have any area.
A **quadrilateral** shape is simply any shape with four sides.