表面積

WordNet

on the surface; "surface materials of the moon"
the outer boundary of an artifact or a material layer constituting or resembling such a boundary; "there is a special cleaner for these surfaces"; "the cloth had a pattern of red dots on a white surface"
the outermost level of the land or sea; "earthquakes originate far below the surface"; "three quarters of the Earths surface is covered by water" (同)Earth''s surface
a superficial aspect as opposed to the real nature of something; "it was not what it appeared to be on the surface"
the extended two-dimensional outer boundary of a three-dimensional object; "they skimmed over the surface of the water"; "a brush small enough to clean every dental surface"; "the sun has no distinct surface"
come to the surface (同)come_up, rise up, rise
the extent of a 2-dimensional surface enclosed within a boundary; "the area of a rectangle"; "it was about 500 square feet in area" (同)expanse, surface_area
a part of a structure having some specific characteristic or function; "the spacious cooking area provided plenty of room for servants"
a particular geographical region of indefinite boundary (usually serving some special purpose or distinguished by its people or culture or geography); "it was a mountainous area"; "Bible country" (同)country
a part of an animal that has a special function or is supplied by a given artery or nerve; "in the abdominal region" (同)region
a subject of study; "it was his area of specialization"; "areas of interest include..."
a unit of surface area equal to 100 square meters (同)ar
emerging to the surface and becoming apparent

PrepTutorEJDIC

(物の)表面,外面;水面 / 《比喩的に》『うわべ』,見かけ / 『表面(水面)の,』表面(水面)に関する / 《比喩的に》『うわべだけの,』見かけだけの / 陸(水)上輸送の / (沈んでいたものが)表面に浮上する;《話》《おどけて》(寝ていた人が)起きる / (…で)…‘に'表面をつける,〈道路〉‘を'舗装する《+名+with+名》
〈U〉〈C〉『面積』 / 〈C〉『地域』,『地方』(region, district) / 〈C〉(活動・研究・興味などの及ぶ)『範囲』,『領域』(range)《+『of』+『名』》 / 〈C〉《英》=areaway 1
…『である』,…です(beの二人称の単数とすべての人称の複数の直説法現在形)
アール(メートル法の面積単位;100平方メートル)
are (面積の単位)の変形

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/07/02 08:42:51」(JST)

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A sphere of radius has surface area

The surface area of a solid object is a measure of the total area that the surface of an 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 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

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

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

with a continuously differentiable function The area of an individual piece is defined by the formula

Thus the area of S_D is obtained by integrating the length of the normal vector 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 that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area (Known as Schwarz's paradox.) ^[1] .^[2]

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		s = side length
Cuboid		ℓ = length, w = width, h = height
Triangular prism		b = base length of triangle, h = height of triangle, l = distance between triangular bases, a, b, c = sides of triangle
All Prisms		B = the area of one base, P = the perimeter of one base, h = height
Sphere		r = radius of sphere, d = diameter
Spherical lune		r = radius of sphere, θ = dihedral angle
Torus		r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus)
Closed cylinder		r = radius of the circular base, h = height of the cylinder
Lateral surface area of a cone		s = slant height of the cone, r = radius of the circular base, h = height of the cone
Full surface area of a cone		s = slant height of the cone, r = radius of the circular base, h = height of the cone
Pyramid		B = area of base, P = perimeter of base, L = slant height
Square pyramid		b = base length, s = slant height, h = vertical height
Rectangular pyramid		ℓ = length, w = width, h = height
Tetrahedron		a = side length

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

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).

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

The discovery of this ratio is credited to Archimedes.^[3]

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, 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).

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.

References

^ http://www.math.usma.edu/people/Rickey/hm/CalcNotes/schwarz-paradox.pdf
^ http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf
^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Retrieved 2007-01-02.

Yu.D. Burago, V.A. Zalgaller, L.D. Kudryavtsev (2001), "Area", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

External links

Surface Area Video at Thinkwell

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