Surface area
This is an accepted version of this page The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies.
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.
This should provide a function which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements.
More rigorously, if a surface S is a union of finitely many pieces S1, …, Sr which do not overlap except at their boundaries, then Surface areas of flat polygonal shapes must agree with their geometrically defined area.
This means that surface area is invariant under the group of Euclidean motions.
Such surfaces consist of finitely many pieces that can be represented in the parametric form with a continuously differentiable function
[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.
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.
= shadow region 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.
For example, iron in a fine powder will combust,[5] while in solid blocks it is stable enough to use in structures.
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.
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)πr3 and SA = 4πr2.
