Main Page: Difference between revisions

Revision as of 03:50, 9 August 2014

The surface area of a solid object is the total area of the object's faces and curved surfaces. 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.

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	$6s^{2}\,$	s = side length
Rectangular prism	$2(\ell w+\ell h+wh)\,$	ℓ = length, w = width, h = height
Triangular prism	$bh+l(a+b+c)$	b = base length of triangle, h = height of triangle, l = distance between triangles, a, b, c = 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
Spherical lune	$2r^{2}\theta \,$	r = radius of sphere, θ = dihedral angle
Torus	$(2\pi r)(2\pi R)=4\pi ^{2}Rr$	r = minor radius, R = major radius
Closed cylinder	$2\pi r^{2}+2\pi rh=2\pi r(r+h)\,$	r = radius of the circular base, h = height of the cylinder
Lateral surface area of a cone	$\pi r\left({\sqrt {r^{2}+h^{2}}}\right)=\pi rs\,$	$s={\sqrt {r^{2}+h^{2}}}$ s = slant height of the cone, r = radius of the circular base, h = height of the cone
Full surface area of a cone	$\pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)=\pi r(r+s)\,$	s = slant height of the cone, r = radius of the circular base, h = height of the cone
Pyramid	$B+{\frac {PL}{2}}$	B = area of base, P = perimeter of base, L = slant height
Square pyramid	$b^{2}+2bs$	b = base length, s = slant height

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.^[3]

In chemistry

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. 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

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.

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

Template:Eom

External links

Surface Area Video at Thinkwell

es:Área de superficies pl:Pole powierzchni sv:Area

[sch1-1] ttp://www.math.usma.edu/people/Rickey/hm/CalcNotes/schwarz-paradox.pdf

[sch2-2] ttp://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf

[3] Template:Cite web

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-{{Aristotelianism |ideas}}
+The '''surface area''' of a solid object is the total area of the object's faces and curved surfaces. 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 [[Face (geometry)|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 surface]]s. This definition of surface area is based on methods of [[infinitesimal]] [[calculus]] and involves [[partial derivative]]s and [[double integration]].
-'''Substance theory''', or '''substance attribute theory''', is an [[ontology|ontological]] theory about [[Object (philosophy)|objecthood]], positing that a ''substance'' is distinct from its [[property (philosophy)|properties]]. A ''thing-in-itself'' is a property-bearer that must be distinguished from the properties it bears.<ref name=Langton>{{cite book |title=Kantian humility: our ignorance of things in themselves |author=Rae Langton |url=http://books.google.com/books?id=DpHUgZzu0EkC&pg=PA28 |page=28 |isbn=0-19-924317-4 |publisher=Oxford University Press |year=2001}}</ref>
+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.
-''Substance'' is a key concept in ontology and [[metaphysics]], which may be classified into [[monist]], [[dualist]] or [[pluralism (philosophy)|pluralist]] varieties according to how many substances or individuals are said to populate, furnish or exist in the world. According to Monistic views, such as those of [[stoicism]] and [[Spinoza]], there is only one substance, [[pneuma]] or [[God]], respectively. These modes of thinking are sometimes associated with the idea of [[immanence]]. Dualism sees the world as being composed of two fundamental substances, for example, the Cartesian [[substance dualism]] of mind and matter. Pluralist philosophies include [[Plato]]'s [[Theory of Forms]] and [[Aristotle]]'s [[hylomorphic]] [[categories (Aristotle)|categories]].
+==Definition==
-==Ancient Greek philosophy==
+While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical ''definition'' of area requires a great deal of care.
-[[Aristotle]] used the term in a secondary sense for [[genus|genera]] and [[species]] understood as [[hylomorphism|hylomorphic forms]]. Primarily, however, he used it with regard to his [[Categories (Aristotle)|category]] of [[substance theory|substance]], the specimen ("this person" or "this horse") or [[identity (philosophy)|individual]], ''qua'' individual, who survives [[accident (philosophy)|accidental change]] and in whom the [[essence|essential properties]] inhere that define those [[problem of universals|universals]]. In contrast, [[Plato]] and later [[Neoplatonism]], spoke of the objective reality of a thing or its [[theory of forms|inner reality]] (as opposed to outer appearance or [[allegory of the cave|illusion]]).
+This should provide a function
-{{quote|A substance—that which is called a substance most strictly, primarily, and most of all—is that which is neither said of a subject nor in a subject, e.g. the individual man or the individual horse. The species in which the things primarily called substances are, are called secondary substances, as also are the genera of these species. For example, the individual man belongs in a species, man, and animal is a genus of the species; so these—both man and animal—are called secondary substances.<ref name="Ackrill1988">{{cite book |first=J.L. |last=Ackrill |year=1988 |title=A New Aristotle Reader |publisher=Princeton University Press |isbn=9781400835829 |url=http://books.google.com/books?id=Cz8-DgxETuAC&pg=PA7 |page=7}}</ref>|Aristotle|''Categories'' 2a13, (trans. J.L. Ackrill)}}
-In chapter 6 of the ''[[Physics]]'' Aristotle argues that any change must be analysed in reference to the property of an invariant subject as it was before the change and thereafter. Thus, in his hylomorphic account of change, ''matter'' serves as a relative substratum of transformation, i.e., of changing form. In the ''Categories'', properties are predicated only of substance, but in chapter 7 of the ''Physics'', Aristotle discusses substances coming to be and passing away in the "unqualified sense" wherein a primary substance is generated from (or perishes into) a material substratum by having gained (or lost) the essential property that formally defines a substance of that kind (in the secondary sense). However, because an essential property remains invariant during an [[accident (philosophy)|accidental]] change in form, by identifying the substance with its formal essence, substance may thereby serve as the relative subject matter or property-bearer of change in a qualified sense (i.e., barring matters of life or death).
+: <math> S \mapsto A(S) </math>
-Neither the "bare particulars" nor "property bundles" of modern theory have their antecedent in Aristotle, according to whom, all matter exists in some form. There is no ''prime matter'' or pure [[classical elements|elements]], there is always a mixture: a ratio weighing the four potential combinations of primary and secondary properties and analysed into discrete one-step and [[gray code#Motivation|two-step]] abstract transmutations between the elements.
+which assigns a positive [[real number]] to a certain class of [[surface]]s 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 ''S''<sub>1</sub>, …, ''S''<sub>''r''</sub> which do not overlap except at their boundaries, then
+: <math> A(S) = A(S_1) + \cdots + A(S_r). </math>
-However, according to [[Aristotelian view of God|Aristotle's theology]], a form of invariant form exists without matter, beyond the [[cosmos]], powerless and oblivious, in the eternal substance of the [[unmoved movers]].
+Surface areas of flat polygonal shapes must agree with their geometrically defined [[area]]. Since surface area is a geometric notion, areas of [[congruence (geometry)|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 [[Euclidean group|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 surface|parametric form]]
-==Early Western philosophy==
+: <math> S_D: \vec{r}=\vec{r}(u,v), \quad (u,v)\in D </math>
-[[Descartes]] means by "substance" an entity which exists in such a way that it needs no other entity in order to exist. Therefore, only God is a substance is the strict sense.  But he extends the term to created things, which need only the concurrence of God to exist. Of these there are two and only two: mind and matter, each being distinct from the other in their attributes and therefore in their essence, and neither needing the other in order to exist. This is Descartes' dualism.  Spinoza denied Descartes' 'real distinction' between mind and matter.  Substance, according to [[Spinoza]], is one and indivisible, but has multiple 'attributes'.  But an 'attribute' is 'what we conceive as constituting the [single] essence of substance'. We may conceive of the single essence of the one substance as material and also, consistently, as mental. What we ordinarily call the natural world, together with all the individuals in it, is [[immanence|immanent]] in God: hence the famous phrase ''deus sive natura'' ("God, or Nature").
-[[John Locke|Locke]] defined substance as follows: {{quote|The idea that we have, to which we give the general name substance, being nothing but the supposed, but unknown, support of those qualities we find existing, which we imagine cannot subsist ''sine re substante'', without something to support them, we call that support ''substantia''; which, according to the true import of the word, is, in plain English, standing under or upholding.|John Locke|"[[An Essay Concerning Human Understanding]]"; book 2, chapter 23; ''Of our Complex Ideas of Substances''}}
+with a [[continuously differentiable]] function <math>\vec{r}.</math> The area of an individual piece is defined by the formula
-==Criticisms of the concept of substance==
+: <math> A(S_D) = \iint_D\left |\vec{r}_u\times\vec{r}_v\right | \, du \, dv. </math>
-{{see also|Noumenon|Phenomenon}}
-The idea of substance was famously critiqued by [[David Hume]],{{Citation needed|date=July 2011}} who held that since substance cannot be perceived, it should not be assumed to exist. But the claim that substance cannot be perceived is neither clear nor obvious, and neither is the implication obvious.
-[[Friedrich Nietzsche]] and, after him, [[Martin Heidegger]], [[Michel Foucault]] and [[Gilles Deleuze]] also rejected the notion of "substance", and in the same movement the concept of [[subject (philosophy)|subject]] contained with the framework of [[Platonic idealism]]. For this reason, [[Althusser]]'s "anti-humanism" and Foucault's statements were criticized, by [[Jürgen Habermas]] and others, for misunderstanding that this led to a fatalist conception of [[social determinism]]. For Habermas, only a subjective form of [[liberty]] could be conceived, to the contrary of Deleuze who talks about "''a'' life", as an impersonal and [[immanence|immanent]] form of liberty.
+Thus the area of ''S''<sub>''D''</sub> is obtained by integrating the length of the normal vector <math>\vec{r}_u\times\vec{r}_v</math> 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 [[surface of revolution|surfaces of revolution]].
-For Heidegger, Descartes means by "substance" that by which "we can understand nothing else than an entity which ''is'' in such a way that it need no other entity in order to ''be''." Therefore, only God is a substance as ''ens perfectissimus'' (most perfect being). Heidegger showed the inextricable relationship between the concept of substance and of subject, which explains why, instead of talking about "man" or "humankind", he speaks about the ''[[Dasein]]'', which is not a simple subject, nor a substance. <ref>{{cite web|url=http://www20.uludag.edu.tr/~kadir/Roma.pdf |author=A. Kadir Cucen |title=Heidegger's Critique of Descartes' Metaphysics |publisher=Uludag University |date=2002-01-18 |accessdate=2011-12-28}}</ref>
+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]].)
+<ref name=sch1>http://www.math.usma.edu/people/Rickey/hm/CalcNotes/schwarz-paradox.pdf</ref>
+.<ref name=sch2>http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf</ref>
-[[Alfred North Whitehead]] has argued that the concept of substance has only a limited applicability in everyday life and that metaphysics should rely upon the concept of process.<ref>See, e.g., Ronny Desmet and Michel Weber (edited by), ''[https://www.academia.edu/279940/Whitehead._The_Algebra_of_Metaphysics Whitehead. The Algebra of Metaphysics. Applied Process Metaphysics Summer Institute Memorandum]'', Louvain-la-Neuve, Éditions Chromatika, 2010 (ISBN 978-2-930517-08-7).</ref>
+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 [[fractal]]s. 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.
-Roman Catholic theologian [[Karl Rahner]], as part of his critique of transubstantiation, rejected substance theory and instead proposed the doctrine of ''transfinalization'', which he felt was more attuned to modern philosophy. However, this doctrine was rejected by Pope [[Paul VI]] in his encyclical ''[[Mysterium fidei (encyclical)|Mysterium fidei]]''.
+== Common formulas ==
-==Irreducible concepts==
+{| class="wikitable"
-Two irreducible concepts encountered in substance theory are the ''bare particular'' and ''inherence''.
+|+ Surface areas of common solids
+|-
+!Shape
+!Equation
+!Variables
+|-
+|[[Cube]]
+|<math> 6s^2 \, </math>
+|''s'' = side length
+|-
+|[[Rectangular prism]]
+|<math> 2(\ell w + \ell h + wh) \, </math>
+|''ℓ'' = length, ''w'' = width, ''h'' = height
+|-
+|[[Triangular prism]]
+|<math> bh + l(a + b + c) </math>
+|''b'' = base length of triangle, ''h'' = height of triangle, ''l'' = distance between triangles, ''a'', ''b'', ''c'' = sides of triangle
+|-
+|All [[Prism (geometry)|Prisms]]
+|<math> 2B + Ph \, </math>
+|''B'' = the area of one base, ''P'' = the perimeter of one base, ''h'' = height
+|-
+|[[Sphere]]
+|<math> 4\pi r^2 = \pi d^2\, </math>
+|''r'' = radius of sphere, ''d'' = diameter
+|-
+|[[Spherical lune]]
+|<math> 2r^2\theta \, </math>
+|''r'' = radius of sphere, ''θ'' = [[dihedral angle]]
+|-
+|[[Torus]]
+|<math> (2\pi r)(2\pi R) = 4\pi^2 Rr</math>
+|''r'' = minor radius, ''R'' = major radius
+|-
+|Closed [[Cylinder (geometry)|cylinder]]
+|<math> 2\pi r^2 + 2\pi rh = 2\pi r(r+h) \, </math>
+|''r'' = radius of the circular base, ''h'' = height of the cylinder
+|-
+|Lateral surface area of a [[cone (geometry)|cone]]
+|<math> \pi r \left(\sqrt{r^2+h^2}\right) = \pi rs \, </math>
+|<math> s = \sqrt{r^2+h^2} </math><br>
+''s'' = slant height of the cone,<br>
+''r'' = radius of the circular base,<br>
+''h'' = height of the cone
+|-
+|Full surface area of a cone
+|<math> \pi r \left(r + \sqrt{r^2+h^2}\right) = \pi r(r + s) \, </math>
+| ''s'' = slant height of the cone,<br>
+''r'' = radius of the circular base,<br>
+''h'' = height of the cone
+|-
+|[[Pyramid (geometry)|Pyramid]]
+|<math>B + \frac{PL}{2}</math>
+|''B'' = area of base, ''P'' = perimeter of base, ''L'' = slant height
+|-
+|[[Square pyramid]]
+|<math> b^2 + 2bs </math>
+|''b'' = base length, ''s'' = slant height
+|}
-===Bare particular===
+===Ratio of surface areas of a sphere and cylinder of the same Radius and Height===
-In substance theory, a bare particular of an [[object (philosophy)|object]] is the element without which the object would not exist, that is, its substance, which exists independently from its properties, even if it is impossible for it to lack properties entirely. It is "bare" because it is considered without its properties and "particular" because it is not [[Abstraction|abstract]]. The properties that the substance has are said to inhere in the substance.
-===Inherence===
+[[Image:Inscribed cone sphere cylinder.svg|thumb|300px|A cone, sphere and cylinder of radius ''r'' and height ''h''.]]
-Another primitive concept in substance theory is the [[inherence]] the properties within a substance. For example, in the sentence, "The apple is red," substance theory says that red inheres in the apple. Substance theory takes the meaning of an apple having the property of redness to be understood, and likewise that of a property's inherence in substance, which is similar to, but not identical with, being part of the substance.
+The below given formulas can be used to show that the surface area of a [[sphere]] and [[cylinder (geometry)|cylinder]] of the same radius and height are in the ratio '''2&nbsp;:&nbsp;3''', as follows.
-The inverse relation is [[Participation (philosophy)|participation]].  Thus in the example above, just as red inheres in the apple, so the apple participates in red.
+Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere).
-==Arguments supporting the theory==
+<math>\begin{array}{rlll}
-Two common [[argument]]s supporting substance theory are the argument from grammar and the argument from conception.
+\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}</math>
-===Argument from grammar===
+The discovery of this ratio is credited to [[Archimedes]].<ref>{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title = Tomb of Archimedes: Sources|publisher = Courant Institute of Mathematical Sciences|accessdate = 2007-01-02}}</ref>
-The argument from grammar uses [[traditional grammar]] to support substance theory. For example, the sentence "Snow is white" contains a grammatical subject "snow" and the predicate "is white", thereby asserting ''snow is white''. The argument holds that it makes no grammatical sense to speak of "whiteness" disembodied, without asserting that snow or something else ''is'' white. Meaningful assertions are formed by virtue of a grammatical subject, of which properties may be predicated, and in substance theory, such assertions are made with regard to a substance.
-[[Bundle theory]] rejects the argument from grammar on the basis that a grammatical subject does not necessarily refer to a metaphysical subject. Bundle theory, for example, maintains that the grammatical subject of statement refers to its properties. For example, a bundle theorist understands the grammatical subject of the sentence, "Snow is white", to be a bundle of properties such as white. Accordingly, one can make meaningful statements about bodies without referring to substances.
+== In chemistry ==
+{{see also|Accessible surface area}}
+Surface area is important in [[chemical kinetics]]. Increasing the surface area of a substance generally increases the [[reaction rate|rate]] of a [[chemical reaction]]. For example, [[iron]] in a fine powder will [[combustion|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.
-===Argument from conception===
+== In biology ==
-Another argument for the substance theory is the argument from conception. The argument claims that in order to conceive of an object's properties, like the redness of an apple, one must conceive of the object that has those properties. According to the argument, one cannot conceive of redness, or any other property, distinct from the substance that has that property.
+{{see also|Surface-area-to-volume ratio}}
+[[Image:Mitochondrion 186.jpg|right|thumb|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. [[Elephant]]s have large [[ear]]s, 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.
-==Bundle theory==
+The [[surface area to volume ratio]] (SA:V) of a [[cell (biology)|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 ''r''<sup>3</sup>; ''SA'' = 4 π ''r''<sup>2</sup>.  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.
-{{Main|Bundle theory}}
-In direct opposition to substance theory is bundle theory, whose most basic premise is that all concrete particulars are merely constructions or 'bundles' of attributes or qualitative properties:
+== References ==
-:Necessarily, for any concrete entity, <math>a</math>, if for any entity, <math>b</math>, <math>b</math> is a constituent of <math>a</math>, then <math>b</math> is an attribute.<ref name="Loux2002"/>
+<references />
-The [[bundle theory|bundle theorist's]] principal objections to substance theory concern the [[bare particular]]s of a substance, which substance theory considers independently of the substance's properties. The bundle theorist objects to the notion of a thing with no properties, claiming that such a thing is inconceivable and citing John Locke, who described a substance as "a something, I know not what." To the bundle theorist, as soon as one has any notion of a substance in mind, a property accompanies that notion.
+* {{eom|title=Area|id=A/a013180|author=Yu.D. Burago, V.A. Zalgaller, L.D. Kudryavtsev}}
-<!--The contents of the foregoing section is also found at [[bare particular]].  Please keep these articles consistent.-->
-===Identity of indiscernibles===
-The [[indiscernible|indiscernibility]] argument from the substance theorist targets those bundle theorists who are also metaphysical realists. Metaphysical realism uses the identity of ''universals'' to compare and identify particulars. Substance theorists say that bundle theory is incompatible with metaphysical realism due to the [[identity of indiscernibles]]: particulars may differ from one another only with respect to their attributes or relations.
-The substance theorist's indiscernibility argument against the metaphysically realistic bundle theorist states that numerically different concrete particulars are discernible from the self-same concrete particular only by virtue of qualitatively different attributes.
-:Necessarily, for any complex objects, <math>a</math> and <math>b</math>, if for any entity, <math>c</math>, <math>c</math> is a constituent of <math>a</math> if and only if <math>c</math> is a constituent of <math>b</math>, then <math>a</math> is numerically identical with <math>b</math>.<ref name="Loux2002"/>
-The indiscernibility argument points out that if bundle theory and discernible concrete particulars theory explain the relationship between attributes, then the identity of indiscernibles theory must also be true:
-:Necessarily, for any concrete objects, <math>a</math> and <math>b</math>, if for any attribute, Φ, Φ is an attribute of <math>a</math> if and only if Φ is an attribute of <math>b</math>, then <math>a</math> is numerically identical with <math>b</math>.<ref name="Loux2002">{{cite book |first=M.J. |last=Loux |year=2002 |title=Metaphysics: A Contemporary Introduction |series=Routledge Contemporary Introductions to Philosophy Series |publisher=Taylor & Francis |isbn=9780415140348 |lccn=97011036 |url=http://books.google.com/books?id=vgxkGIgPlooC&pg=PA106 |page=106-107,110}}</ref>
-The indiscernibles argument then asserts that the identity of indiscernibles is violated, for example, by identical sheets of paper. All of their qualitative properties are the same (e.g. white, rectangular, 9 x 11&nbsp;inches...) and thus, the argument claims, bundle theory and metaphysical realism cannot both be correct.
-However, bundle theory combined with [[Trope (philosophy)#Trope theory in philosophy (metaphysics)|trope theory]] (as opposed to metaphysical realism) avoids the indiscernibles argument because each attribute is a trope if can only be held by only one concrete particular.
-The argument does not consider whether "position" should be considered an attribute or relation. It is after all through the differing positions that we in practice differentiate between otherwise identical pieces of paper.
-==Stoicism==
-The [[Stoicism|Stoics]] rejected the idea that [[incorporeal]] beings inhere in matter, as taught by [[Plato]]. They believed that all being is [[Matter|corporeal]] infused with a creative fire called [[pneuma]]. Thus they developed a scheme of [[Categories (Stoic)|categories]] different from [[Categories (Aristotle)|Aristotle's]] based on the ideas of [[Anaxagoras]] and [[Timaeus (dialogue)|Timaeus]].
-==See also==
-{{multicol}}
-*[[Bundle theory]]
-*[[Categories (Stoic)]]
-*[[Dualism]]
-*[[History of chemistry]]
-*[[History of molecular theory]]
-*[[Hyle]]
-{{multicol-break}}
-*[[Hypokeimenon]]
-*[[Hypostasis (philosophy and religion)]]
-*[[Inherence]]
-*[[Materialism]]
-*[[Matter]]
-*[[Metaphysics]]
-{{multicol-break}}
-*[[Monism]]
-*[[Ontology]]
-*[[Ousia]]
-*[[Physical ontology]]
-*[[Process philosophy]]
-*[[Trope (philosophy)]]
-*[[Universals]]
-{{multicol-end}}
-==References==
-{{refimprove|date=October 2010}}
-<references/>
 ==External links==
-* {{sep entry|substance|Substance|Howard Robinson}}
+*[http://blog.thinkwell.com/2010/07/6th-grade-math-surface-area.html Surface Area Video] at Thinkwell
-* [http://www.friesian.com/essence.htm Friesian School on Substance and Essence]
-{{metaphysics}}
+[[Category:Area]]
-{{DEFAULTSORT:Substance Theory}}
+[[es:Área de superficies]]
-[[Category:Ontology]]
+[[pl:Pole powierzchni]]
-[[Category:Aristotelianism]]
+[[sv:Area]]
-[[Category:Metaphysical theories]]

Main Page: Difference between revisions

Revision as of 03:50, 9 August 2014

Contents

Definition

Common formulas

Ratio of surface areas of a sphere and cylinder of the same Radius and Height

In chemistry

In biology

References

External links

Navigation menu

Main Page: Difference between revisions

Revision as of 03:50, 9 August 2014

Definition

Common formulas

Ratio of surface areas of a sphere and cylinder of the same Radius and Height

In chemistry

In biology

References

External links

Navigation menu

Search