The [url=https://en.wikipedia.org/wiki/Surface_area]surface area[/url] of a sphere of radius r is:[br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/95625828c519b36791d56af65d21b8448472650d[/img][url=https://en.wikipedia.org/wiki/Archimedes]Archimedes[/url] first derived this formula from the fact that the projection to the lateral surface of a [url=https://en.wikipedia.org/wiki/Circumscribe]circumscribed[/url] cylinder is area-preserving.Another approach to obtaining the formula comes from the fact that it equals the [url=https://en.wikipedia.org/wiki/Derivative]derivative[/url] of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness. At any given radius r,the incremental volume (δV) equals the product of the surface area at radius r ([i]A[/i]([i]r[/i])) and the thickness of a shell (δr):[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd6b20b042b7b8356514cafe4bfe9323f73970e[/img]The total volume is the summation of all shell volumes:[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff662cf129a7b7ae53bb28460dadef0715b60c4[/img][br]In the limit as δr approaches zero[sup][url=https://en.wikipedia.org/wiki/Sphere#cite_note-delta-8][8][/url][/sup] this equation becomes:[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/dde5072ef2871126b29c8eab1cc7b83ec2d365ad[/img][br]Substitute V:[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/521cb0af0bc79aaa0a4e3c6d6f02d440057fb894[/img][br]Differentiating both sides of this equation with respect to r yields A as a function of r:{\displaystyle 4\pi r^{2}=A(r).}[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/f25b8b4e3cc58f767579a25d212571f323e2f740[/img]This is generally abbreviated as:{\displaystyle A=4\pi r^{2},}[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1cbdbb8a22c7db3b3f01dad4ea19f8dfcd502b[/img]where r is now considered to be the fixed radius of the sphere. Alternatively, the [url=https://en.wikipedia.org/wiki/Area_element]area element[/url] on the sphere is given in [url=https://en.wikipedia.org/wiki/Spherical_coordinates]spherical coordinates[/url] by [i]dA[/i] = [i]r[/i][sup]2[/sup] sin [i]θ dθ dφ[/i]. In [url=https://en.wikipedia.org/wiki/Cartesian_coordinates]Cartesian coordinates[/url], the area element is:[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/78277b9aae2fc2a99ab712de135bfefe1aba6e1d[/img][br]The total area can thus be obtained by [url=https://en.wikipedia.org/wiki/Integral]integration[/url]:[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/128b56cc8737351056a8e5fd4dfc2dd163f58bc5[/img][br]The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere, therefore, appears in nature: for example, bubbles and small water drops are roughly spherical because the [url=https://en.wikipedia.org/wiki/Surface_tension]surface tension[/url] locally minimizes surface area. The surface area relative to the mass of a ball is called the [url=https://en.wikipedia.org/wiki/Specific_surface_area]specific surface area[/url] and can be expressed from the above-stated equations as:[br][br][img]https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6ff5a4a2a0552ea9001c624b565762f2d75510[/img][br]where ρ is the [url=https://en.wikipedia.org/wiki/Density]density[br][br][/url]