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Surface area

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Measure of a two-dimensional surface

Asphere of radiusr has surface area4πr².

Thesurface area (symbolA) of asolid object is a measure of the totalarea that thesurface of the object occupies.^[1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition ofarc length of one-dimensional curves, or of the surface area forpolyhedra (i.e., objects with flat polygonalfaces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as asphere, are assigned surface area using their representation asparametric surfaces. This definition of surface area is based on methods ofinfinitesimal calculus and involvespartial derivatives anddouble integration.

A general definition of surface area was sought byHenri Lebesgue andHermann Minkowski at the turn of the twentieth century. Their work led to the development ofgeometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is theMinkowski content of a surface.

Definition

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While the areas of many simple surfaces have been known since antiquity, a rigorous mathematicaldefinition of area requires a great deal of care.This should provide a function

S\mapsto A(S)

which assigns a positivereal number to a certain class ofsurfaces that satisfies several natural requirements. The most fundamental property of the surface area is itsadditivity:the area of the whole is the sum of the areas of the parts. More rigorously, if a surfaceS is a union of finitely many piecesS₁, …,S_r which do not overlap except at their boundaries, then

A(S)=A(S_{1})+\cdots +A(S_{r}).

Surface areas of flat polygonal shapes must agree with their geometrically definedarea. Since surface area is a geometric notion, areas ofcongruent surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under thegroup of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces calledpiecewise smooth. Such surfaces consist of finitely many pieces that can be represented in theparametric form

S_{D}:{\vec {r}}={\vec {r}}(u,v),\quad (u,v)\in D

with acontinuously differentiable function ${\vec {r}}.$ The area of an individual piece is defined by the formula

A(S_{D})=\iint _{D}\left|{\vec {r}}_{u}\times {\vec {r}}_{v}\right|\,du\,dv.

Thus the area ofS_D is obtained by integrating the length of the normal vector ${\vec {r}}_{u}\times {\vec {r}}_{v}$ to the surface over the appropriate regionD in the parametricuv plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphsz =f(x,y) andsurfaces of revolution.

Schwarz lantern with $M {\displaystyle M}$ axial slices and $N {\displaystyle N}$ radial vertices. The limit of the area as $M {\displaystyle M}$ and $N {\displaystyle N}$ tend to infinity doesn't converge. In particular it doesn't converge to the area of the cylinder.

Schwarz lantern with $M {\displaystyle M}$ axial slices and $N {\displaystyle N}$ radial vertices. The limit of the area as $M {\displaystyle M}$ and $N {\displaystyle N}$ tend to infinity doesn't converge. In particular it doesn't converge to the area of the cylinder.

One of the subtleties of surface area, as compared toarc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated byHermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as theSchwarz lantern.^[2]^[3]

Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century byHenri Lebesgue andHermann Minkowski. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study offractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied ingeometric measure theory. A specific example of such an extension is theMinkowski content of the surface.

Common formulas

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Surface areas of common solids
Shape	Formula/Equation	Variables
Cube	$6a^{2}$	a = side length
Cuboid	$2\left(lb+lh+bh\right)$	l = length,b = breadth,h = height
Triangular prism	$bh+l\left(p+q+r\right)$	b = base length of triangle,h = height of triangle,l = distance between triangular bases,p,q,r = sides of triangle
Allprisms	$2B+Ph$	B = the area of one base,P = the perimeter of one base,h = height
Sphere	$4\pi r^{2}=\pi d^{2}$	r = radius of sphere,d = diameter
Hemisphere	$3\pi r^{2}$	r = radius of the hemisphere
Hemispherical shell	$\pi \left(3R^{2}+r^{2}\right)$	R = external radius of hemisphere,r = internal radius of hemisphere
Spherical lune	$2r^{2}\theta$	r = radius of sphere,θ =dihedral angle
Torus	$\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr$	r = minor radius (radius of the tube),R = major radius (distance from center of tube to center of torus)
Closedcylinder	$2\pi r^{2}+2\pi rh=2\pi r\left(r+h\right)$	r = radius of the circular base,h = height of the cylinder
Cylindricalannulus	$2\pi Rh+2\pi rh+2(\pi R^{2}-\pi r^{2})=2\pi (R+r)(R-r+h)$	R = External radius r = Internal radius,h = height
Capsule	$2\pi r(2r+h)$	r = radius of the hemispheres and cylinder,h = height of the cylinder
Curved surface area of acone	$\pi r{\sqrt {r^{2}+h^{2}}}=\pi rs$	$s={\sqrt {r^{2}+h^{2}}}$ s = slant height of the cone,r = radius of the circular base,h = height of the cone
Full surface area of a cone	$\pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)=\pi r\left(r+s\right)$	s = slant height of the cone,r = radius of the circular base,h = height of the cone
RegularPyramid	$B+{\frac {Ps}{2}}$	B = area of base,P = perimeter of base,s = slant height
Square pyramid	$b^{2}+2bs=b^{2}+2b{\sqrt {\left({\frac {b}{2}}\right)^{2}+h^{2}}}$	b = base length,s = slant height,h = vertical height
Rectangular pyramid	$lb+l{\sqrt {\left({\frac {b}{2}}\right)^{2}+h^{2}}}+b{\sqrt {\left({\frac {l}{2}}\right)^{2}+h^{2}}}$	l = length,b = breadth,h = height
Tetrahedron	${\sqrt {3}}a^{2}$	a = side length
Surface of revolution	$2\pi \int _{a}^{b}{f(x){\sqrt {1+(f'(x))^{2}}}dx}$
Parametric surface	$\iint _{D}\left\vert {\vec {r}}_{u}\times {\vec {r}}_{v}\right\vert dA$	${\vec {r}}$ = parametric vector equation of surface, ${\vec {r}}_{u}$ = partial derivative of ${\vec {r}}$ with respect to $u {\displaystyle u}$ , ${\vec {r}}_{v}$ = partial derivative of ${\vec {r}}$ with respect to $v {\displaystyle v}$ , $D {\displaystyle D}$ = shadow region

Ratio of surface areas of a sphere and cylinder of the same radius and height

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A cone, sphere and cylinder of radiusr and heighth.

The below given formulas can be used to show that the surface area of asphere andcylinder of the same radius and height are in the ratio2 : 3, as follows.

Let the radius ber and the height beh (which is 2r for the sphere).

${\begin{array}{rlll}{\text{Sphere surface area}}&=4\pi r^{2}&&=(2\pi r^{2})\times 2\\{\text{Cylinder surface area}}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3\end{array}}$

The discovery of this ratio is credited toArchimedes.^[4]

In chemistry

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Surface area of particles of different sizes.

In biology

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Theinner membrane of the mitochondrion has a large surface area due to infoldings, allowing higher rates ofcellular respiration (electronmicrograph).^[6]

The surface area of an organism is important in several considerations, such as regulation of body temperature anddigestion.^[7] Animals use theirteeth to grind food down into smaller particles, increasing the surface area available for digestion.^[8] The epithelial tissue lining the digestive tract containsmicrovilli, greatly increasing the area available for absorption.^[9]Elephants have largeears, allowing them to regulate their own body temperature.^[10] In other instances, animals will need to minimize surface area;^[11] for example, people will fold their arms over their chest when cold to minimize heat loss.

Thesurface area to volume ratio (SA:V) of acell imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across thecell membrane to interstitial spaces or to other cells.^[12] Indeed, representing a cell as an idealizedsphere of radiusr, the volume and surface area are, respectively,V = (4/3)πr³ andSA = 4πr². The resulting surface area to volume ratio is therefore3/r. Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume.