Surface area

Last updated April 01, 2024

The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies.^[1] 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 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 surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives 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.

Definition

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

S\mapsto A(S)

which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface S is a union of finitely many pieces S₁, …, 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 defined area. Since surface area is a geometric notion, areas of congruent 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 the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of finitely many pieces that can be represented in the parametric form

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

with a continuously 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 of S_D is obtained by integrating the length of the normal vector ${\vec {r}}_{u}\times {\vec {r}}_{v}$ to the surface over the appropriate region D in the parametric uv 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 graphs z = f(x,y) and surfaces 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.gif — Schwarz lantern with $M$ axial slices and $N$ radial vertices. The limit of the area as $M$ and $N$ 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 to arc 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 by Hermann 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 the Schwarz lantern.^[2]^[3]

Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann 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 of fractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory. A specific example of such an extension is the Minkowski content of the surface.

Common formulas

Surface areas of common solids
Shape	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
All prisms	$2B+Ph$	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)
Closed cylinder	$2\pi r^{2}+2\pi rh=2\pi r\left(r+h\right)$	r = radius of the circular base, h = height of the cylinder
Cylindrical annulus	$2\pi Rh+2\pi rh+2(\pi R^{2}-\pi r^{2})=2\pi (R+r)(R-r+h)$	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 a cone	$\pi r{\sqrt {r^{2}+h^{2}}}=\pi rs$	$s={\sqrt {r^{2}+h^{2}}}$ 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
Regular Pyramid	$B+{\frac {Ps}{2}}$	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 $u$ , ${\vec {r}}_{v}$ = partial derivative of ${\vec {r}}$ with respect to $v$ , $D$ = shadow region

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

The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows.

Let the radius be r and the height be h (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 to Archimedes.^[4]

In chemistry

Surface area is important in chemical kinetics. Increasing the surface area of a substance generally increases the rate of a chemical reaction. For example, iron in a fine powder will combust,^[5] while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.

In biology

The inner membrane of the mitochondrion has a large surface area due to infoldings, allowing higher rates of cellular respiration (electron micrograph). Mitochondrion 186.jpg — The inner membrane of the mitochondrion has a large surface area due to infoldings, allowing higher rates of cellular respiration (electron micrograph).

The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion.^[7] Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion.^[8] The epithelial tissue lining the digestive tract contains microvilli, greatly increasing the area available for absorption.^[9] Elephants have large ears, 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.

The surface area to volume ratio (SA:V) of a cell 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 the cell membrane to interstitial spaces or to other cells.^[12] Indeed, representing a cell as an idealized sphere of radius $r$ , the volume and surface area are, respectively, $V = (4/3) πr 3$ and $SA = 4 πr 2$ . The resulting surface area to volume ratio is therefore $3/ r$ . 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.

Related Research Articles

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid . Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity $, a number ranging from to .$

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.

Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function called the taxicab distance, Manhattan distance, or city block distance. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length.

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

<span class="mw-page-title-main">Cylinder</span> Three-dimensional solid

A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.

<span class="mw-page-title-main">Parallel curve</span>

A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points are at a constant normal distance from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not.

<span class="mw-page-title-main">Radius</span> Segment in a circle or sphere from its center to its perimeter or surface and its length

In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin radius, meaning ray but also the spoke of a chariot wheel. The typical abbreviation and mathematical variable name for radius is R or r. By extension, the diameter D is defined as twice the radius:

A right circular cylinder is a cylinder whose generatrices are perpendicular to the bases. Thus, in a right circular cylinder, the generatrix and the height have the same measurements. It is also less often called a cylinder of revolution, because it can be obtained by rotating a rectangle of sides $and around one of its sides. Fixing as the side on which the revolution takes place, we obtain that the side, perpendicular to, will be the measure of the radius of the cylinder.$

In geometry, the area enclosed by a circle of radius $r$ is $π r 2$ . Here the Greek letter $π$ represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

A parametric surface is a surface in the Euclidean space $which is defined by a parametric equation with two parameters :\mathbb {R} ^{2}\to \mathbb {R} ^{3}} . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.$

The surface-area-to-volume ratio or surface-to-volume ratio is the ratio between surface area and volume of an object or collection of objects.

On the Sphere and Cylinder is a treatise that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.

In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune. The angle between the radii lying within the bounding semidisks is the dihedral $α$ . If $AB$ is a semidisk that forms a ball when completely revolved about the z-axis, revolving $AB$ only through a given $α$ produces a spherical wedge of the same angle $α$ . Beman (2008) remarks that "a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon." A spherical wedge of $α = π$ radians (180°) is called a hemisphere, while a spherical wedge of $α = 2 π$ radians (360°) constitutes a complete ball.

In geometry, a mylar balloon is a surface of revolution. While a sphere is the surface that encloses a maximal volume for a given surface area, the mylar balloon instead maximizes volume for a given generatrix arc length. It resembles a slightly flattened sphere.

In Einstein's theory of general relativity, the interior Schwarzschild metric is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid and has zero pressure at the surface. This is a static solution, meaning that it does not change over time. It was discovered by Karl Schwarzschild in 1916, who earlier had found the exterior Schwarzschild metric.

In mathematics, the theory of finite sphere packing concerns the question of how a finite number of equally-sized spheres can be most efficiently packed. The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid by László Fejes Tóth.

References

↑ Weisstein, Eric W. "Surface Area". MathWorld .
↑ "Schwarz's Paradox" (PDF). Archived (PDF) from the original on 4 March 2016. Retrieved 21 March 2017.
↑ "Archived copy" (PDF). Archived from the original (PDF) on 15 December 2011. Retrieved 24 July 2012.{{cite web}}: CS1 maint: archived copy as title (link)
↑ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Archived from the original on 9 December 2006. Retrieved 2 January 2007.
↑ Nasr, Somaye; Plucknett, Kevin P. (20 February 2014). "Kinetics of Iron Ore Reduction by Methane for Chemical Looping Combustion". Energy & Fuels. 28 (2): 1387–1395. doi:10.1021/ef402142q. ISSN 0887-0624.
↑ Paumard, Patrick; Vaillier, Jacques; Coulary, Bénédicte; Schaeffer, Jacques; Soubannier, Vincent; Mueller, David M.; Brèthes, Daniel; di Rago, Jean-Paul; Velours, Jean (1 February 2002). "The ATP synthase is involved in generating mitochondrial cristae morphology". The EMBO Journal. 21 (3): 221–230. doi:10.1093/emboj/21.3.221. PMC 125827 . PMID 11823415.
↑ Narasimhan, Arunn (1 July 2008). "Why do elephants have big ear flaps?". Resonance. 13 (7): 638–647. doi:10.1007/s12045-008-0070-5. ISSN 0973-712X.
↑ Feher, Joseph (2012), "Mouth and Esophagus", Quantitative Human Physiology, Elsevier, pp. 689–700, doi:10.1016/b978-0-12-382163-8.00077-3, ISBN 978-0-12-382163-8 , retrieved 30 March 2024
↑ "Microvillus | Description, Anatomy, & Function | Britannica". www.britannica.com. Retrieved 30 March 2024.
↑ Wright, P. G. (1984). "Why do elephants flap their ears?". African Zoology. 19 (4): 266–269. ISSN 2224-073X.
↑ Stocks, Jodie M.; Taylor, Nigel A.S.; Tipton, Michael J.; Greenleaf, John E. (1 May 2004). "Human Physiological Responses to Cold Exposure". Aviation, Space, and Environmental Medicine. 75 (5): 444–457. PMID 15152898.
↑ Deaver, James R. (1 November 1978). "Modeling Limits to Cell Size". The American Biology Teacher. 40 (8): 502–504. doi:10.2307/4446369. ISSN 0002-7685. JSTOR 4446369.

Yu.D. Burago; V.A. Zalgaller; L.D. Kudryavtsev (2001) [1994], "Area", Encyclopedia of Mathematics , EMS Press

External links

Surface Area Video at Thinkwell

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[1] Weisstein, Eric W. "Surface Area". MathWorld .

[sch1-2] "Schwarz's Paradox" (PDF). Archived (PDF) from the original on 4 March 2016. Retrieved 21 March 2017.

[sch2-3] "Archived copy" (PDF). Archived from the original (PDF) on 15 December 2011. Retrieved 24 July 2012.{{cite web}}: CS1 maint: archived copy as title (link)

[4] Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Archived from the original on 9 December 2006. Retrieved 2 January 2007.

[5] Nasr, Somaye; Plucknett, Kevin P. (20 February 2014). "Kinetics of Iron Ore Reduction by Methane for Chemical Looping Combustion". Energy & Fuels. 28 (2): 1387–1395. doi:10.1021/ef402142q. ISSN 0887-0624.

[6] Paumard, Patrick; Vaillier, Jacques; Coulary, Bénédicte; Schaeffer, Jacques; Soubannier, Vincent; Mueller, David M.; Brèthes, Daniel; di Rago, Jean-Paul; Velours, Jean (1 February 2002). "The ATP synthase is involved in generating mitochondrial cristae morphology". The EMBO Journal. 21 (3): 221–230. doi:10.1093/emboj/21.3.221. PMC 125827 . PMID 11823415.

[7] Narasimhan, Arunn (1 July 2008). "Why do elephants have big ear flaps?". Resonance. 13 (7): 638–647. doi:10.1007/s12045-008-0070-5. ISSN 0973-712X.

[8] Feher, Joseph (2012), "Mouth and Esophagus", Quantitative Human Physiology, Elsevier, pp. 689–700, doi:10.1016/b978-0-12-382163-8.00077-3, ISBN 978-0-12-382163-8 , retrieved 30 March 2024

[9] "Microvillus | Description, Anatomy, & Function | Britannica". www.britannica.com. Retrieved 30 March 2024.

[10] Wright, P. G. (1984). "Why do elephants flap their ears?". African Zoology. 19 (4): 266–269. ISSN 2224-073X.

[11] Stocks, Jodie M.; Taylor, Nigel A.S.; Tipton, Michael J.; Greenleaf, John E. (1 May 2004). "Human Physiological Responses to Cold Exposure". Aviation, Space, and Environmental Medicine. 75 (5): 444–457. PMID 15152898.

[12] Deaver, James R. (1 November 1978). "Modeling Limits to Cell Size". The American Biology Teacher. 40 (8): 502–504. doi:10.2307/4446369. ISSN 0002-7685. JSTOR 4446369.

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