Editing Area (section)

===Non-planar surface area===
{{main article|Surface area}}
[[File:Archimedes sphere and cylinder.svg|right|thumb|alt=A blue sphere inside a cylinder of the same height and radius|[[Archimedes]] showed that the surface area of a [[sphere]] is exactly four times the area of a flat [[disk (mathematics)|disk]] of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a [[cylinder (geometry)|cylinder]] of the same height and radius.]]

Most basic formulas for [[surface area]] can be obtained by cutting surfaces and flattening them out (see: [[developable surface]]s).  For example, if the side surface of a [[cylinder (geometry)|cylinder]] (or any [[prism (geometry)|prism]]) is cut lengthwise, the surface can be flattened out into a rectangle.  Similarly, if a cut is made along the side of a [[cone (geometry)|cone]], the side surface can be flattened out into a sector of a circle, and the resulting area computed.

The formula for the surface area of a [[sphere]] is more difficult to derive: because a sphere has nonzero [[Gaussian curvature]], it cannot be flattened out.  The formula for the surface area of a sphere was first obtained by [[Archimedes]] in his work ''[[On the Sphere and Cylinder]]''.  The formula is:<ref name=MathWorldSurfaceArea/>
:{{math|''A'' {{=}} 4''πr''<sup>2</sup>}} &nbsp;(sphere), 
where {{math|''r''}} is the radius of the sphere.  As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to [[calculus]].