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 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
↦
A
(
S
)
{\displaystyle 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, …, 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 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.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 ==
=== 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).
Sphere surface area
=
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
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.
== 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 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 ==
Perimeter length
BET theory, technique for the measurement
of the specific surface area of materials
Spherical area
Surface integral
