Editing Surface area (section)

{{Short description|Measure of a two-dimensional surface}}
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[[Image:Sphere wireframe 10deg 6r.svg|right|thumb|A [[sphere]] of radius {{mvar|r}} has surface area {{math|4''&pi;r''<sup>2</sup>}}.]]

The '''surface area''' (symbol '''''A''''') of a [[Solid geometry|solid]] object is a measure of the total [[area]] that the [[Surface (mathematics)|surface]] of the object occupies.<ref>{{MathWorld|title=Surface Area|urlname=SurfaceArea}}</ref> The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of [[arc length]] of one-dimensional curves, or of the surface area for [[polyhedra]] (i.e., objects with flat polygonal [[Face (geometry)|faces]]), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a [[sphere]], are assigned surface area using their representation as [[parametric surface]]s. This definition of surface area is based on methods of [[infinitesimal calculus]] and involves [[partial derivative]]s and [[double integration]].

A general definition of surface area was sought by [[Henri Lebesgue]] and [[Hermann Minkowski]] at the turn of the twentieth century. Their work led to the development of [[geometric measure theory]], which studies various notions of surface area for irregular objects of any dimension. An important example is the [[Minkowski content]] of a surface.