Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of surfaces.
Units for measuring area include:
All of the above calculations show how to find the area of many shapes.
The area of irregular polygons can be calculated using the "Surveyor's formula".[1]
Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area, cf. space-filling curves. A figure may have infinite area, for example the entire Euclidean plane. The 3-dimensional analog of area is the volume. Although area seems to be one of the basic notions in geometry, it is not easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. For polygons in the Euclidean plane, one can proceed as follows:
It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts.
A typical way to introduce area is through the more advanced notion of Lebesgue measure. In the presence of the axiom of choice it is possible to prove the existence of shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach–Tarski paradox). The sets involved do not arise in practical matters.
In three dimensions, the analog of area is called volume. The n dimensional analog, usually referred to as 'content', is defined by means of a measure or as a Lebesgue integral.
(see Green's theorem)
The general formula for the surface area of the graph of a continuously differentiable function z = f(x,y), where 
and D is a region in the xy-plane with the smooth boundary:
Even more general formula for the area of the graph of a parametric surface in the vector form 
where 
is a continuously differentiable vector function of 
:
Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.
The question of the filling area of the Riemannian circle remains open.
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