The surface area of a solid object is a measure of the total area that
the surface of the object occupies. The mathematical definition of
surface area in the presence of curved surfaces is considerably more
involved than the definition of arc length of onedimensional 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
Contents 1 Definition 2 Common formulas 2.1 Ratio of surface areas of a sphere and cylinder of the same radius and height 3 In chemistry 4 In biology 5 See also 6 References 7 External links Definition[edit] 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 ↦ A ( S ) displaystyle Smapsto 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, …, Sr which do not overlap except at their boundaries, then A ( S ) = A ( S 1 ) + ⋯ + A ( S r ) . displaystyle 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 : r → = r → ( u , v ) , ( u , v ) ∈ D displaystyle S_ D : vec r = vec r (u,v),quad (u,v)in D with a continuously differentiable function r → . displaystyle vec r . The area of an individual piece is defined by the formula A ( S D ) = ∬ D
r → u × r → v
d u d v . displaystyle A(S_ D )=iint _ D left vec r _ u times vec r _ v right,du,dv. Thus the area of SD is obtained by integrating the length of the normal vector r → u × r → v displaystyle 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.
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
Surface areas of common solids Shape Equation Variables Cube 6 s 2 displaystyle 6s^ 2 , s = side length Cuboid 2 ( ℓ w + ℓ h + w h ) displaystyle 2(ell w+ell h+wh), ℓ = length, w = width, h = height Triangular prism b h + l ( a + b + c ) displaystyle bh+l(a+b+c) b = base length of triangle, h = height of triangle, l = distance between triangular bases, a, b, c = sides of triangle All prisms 2 B + P h displaystyle 2B+Ph, B = the area of one base, P = the perimeter of one base, h = height Sphere 4 π r 2 = π d 2 displaystyle 4pi r^ 2 =pi d^ 2 , r = radius of sphere, d = diameter Spherical lune 2 r 2 θ displaystyle 2r^ 2 theta , r = radius of sphere, θ = dihedral angle Torus ( 2 π r ) ( 2 π R ) = 4 π 2 R r displaystyle (2pi r)(2pi R)=4pi ^ 2 Rr r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus) Closed cylinder 2 π r 2 + 2 π r h = 2 π r ( r + h ) displaystyle 2pi r^ 2 +2pi rh=2pi r(r+h), r = radius of the circular base, h = height of the cylinder Lateral surface area of a cone π r ( r 2 + h 2 ) = π r s displaystyle pi rleft( sqrt r^ 2 +h^ 2 right)=pi rs, s = r 2 + h 2 displaystyle 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 π r ( r + r 2 + h 2 ) = π r ( r + s ) displaystyle pi rleft(r+ sqrt r^ 2 +h^ 2 right)=pi r(r+s), s = slant height of the cone, r = radius of the circular base, h = height of the cone Pyramid B + P L 2 displaystyle B+ frac PL 2 B = area of base, P = perimeter of base, L = slant height Square pyramid b 2 + 2 b s = b 2 + 2 b ( b 2 ) 2 + h 2 displaystyle 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 l w + l ( w 2 ) 2 + h 2 + w ( l 2 ) 2 + h 2 displaystyle lw+l sqrt left( frac w 2 right)^ 2 +h^ 2 +w sqrt left( frac l 2 right)^ 2 +h^ 2 ℓ = length, w = width, h = height Tetrahedron 3 a 2 displaystyle sqrt 3 a^ 2 a = side length Ratio of surface areas of a sphere and cylinder of the same radius and height[edit] A cone, sphere and cylinder of radius r and height h. 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).
Sphere
= 4 π r 2 = ( 2 π r 2 ) × 2 Cylinder surface area = 2 π r ( h + r ) = 2 π r ( 2 r + r ) = ( 2 π r 2 ) × 3 displaystyle begin array rlll text
Sphere
The discovery of this ratio is credited to Archimedes.[3] In chemistry[edit]
Surface area
See also: Accessible surface area
Surface area
In biology[edit] See also: Surfaceareatovolume ratio 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. Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains microvilli, greatly increasing the area available for absorption. Elephants have large ears, allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; 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. Indeed, representing a cell as an idealized sphere of radius r, the volume and surface area are, respectively, V = 4/3 π r3; SA = 4 π r2. 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. See also[edit] Perimeter length BET theory, technique for the measurement of the specific surface area of materials Surface integral References[edit] ^ (PDF) http://fredrickey.info/hm/CalcNotes/schwarzparadox.pdf. Archived (PDF) from the original on 20160304. Retrieved 20170321. Missing or empty title= (help) ^ "Archived copy" (PDF). Archived from the original (PDF) on 20111215. Retrieved 20120724. ^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Archived from the original on 20061209. Retrieved 20070102. Yu.D. Burago, V.A. Zalgaller, L.D. Kudryavtsev (2001) [1994], "Area", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104 CS1 maint: Multiple names: authors list (link) External links[edit] Surface
Area
