Surface area

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is the measure of how much exposed area an object has. It is expressed in square units. If an object has flat faces, its surface area can be calculated by adding together the areas of its faces. Even objects with smooth surfaces, such as a sphere has surface area

Formulas

Common equations for surface area (3-Dimensional Objects):
Shape	Equation	Variables
A cube:	$6 cdot s^2$	s = length of any side
A rectangular prism:	$2 cdot (l cdot w + l cdot h + w cdot h)$	l = length, w = width, h = height
A sphere:	$4 cdot pi cdot r^2$	r = radius of sphere, which is the first derivative of the volume of a sphere
A cylinder:	$2 cdot pi cdot r cdot (h+r)$	r' = radius of circular base, h'' = height
A cone (lateral surface area):	$pi cdot r cdot [(r + sqrt{(r^2+h^2)}]$	r = radius of circular base, "h" = height
A cone:	$pi cdot r^2 + pi cdot rcdot s$	r = radius of circular base, s = slant height of the cone

Shape	Area formula derivation
Sphere	The surface area of a sphere is the integral of infinitesimal circular rings of width $dx$ The radius of the circular ring is $f(x) = sqrt(r^2-x^2)$ . The length of the circular ring is equal to $2picdot f(x)$ The width of the ring can be determined by using Pythagoras' formula for a rectangular triangle with side lengths $dx$ and $f'(x) cdot dx$ , which leads to $sqrt(1+f'(x)^2)dx$ The infinitesimal surface area of the circular ring thus is equal to $2pi f(x)cdot sqrt(1+f'(x)^2)dx$ The derivative of $f(x)$ is equal to $f'(x) = frac{-x}{sqrt(r^2-x^2)}$ The surface area of the sphere can be calculated as $int_{-r}^r 2pi f(x)cdot sqrt(1+f'(x)^2),dx$ = $int_{-r}^r 2pi sqrt(r^2-x^2) cdot sqrt(1+frac{x^2}{r^2-x^2}),dx = int_{-r}^r 2pi sqrt {r^2},dx = 2pi r int_{-r}^r 1,dx$ The antiderivative needed is the simple linear function $x$ Thus, the sphere surface area amounts to A_sphere = $2pi r[r-(-r)] = 4pi r^2$

Shape

Area formula derivation

Sphere

The surface area of a sphere is the integral of infinitesimal circular rings of width

dx

The radius of the circular ring is

f(x) = sqrt(r^2-x^2)

. The length of the circular ring is equal to

2picdot f(x)

The width of the ring can be determined by using Pythagoras' formula for a rectangular triangle with side lengths

dx

and

f'(x) cdot dx

, which leads to

sqrt(1+f'(x)^2)dx

The infinitesimal surface area of the circular ring thus is equal to

2pi f(x)cdot sqrt(1+f'(x)^2)dx

The derivative of

f(x)

is equal to

f'(x) = frac{-x}{sqrt(r^2-x^2)}

The surface area of the sphere can be calculated as

int_{-r}^r 2pi f(x)cdot sqrt(1+f'(x)^2),dx

int_{-r}^r 2pi sqrt(r^2-x^2) cdot sqrt(1+frac{x^2}{r^2-x^2}),dx = int_{-r}^r 2pi sqrt {r^2},dx = 2pi r int_{-r}^r 1,dx

The antiderivative needed is the simple linear function

x

Thus, the sphere surface area amounts to

A_sphere =

2pi r[r-(-r)] = 4pi r^2

Surfaces whose area cannot be defined

While areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a lot of care. Various approaches to defining the surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. For a very wide class of geometric surfaces called piecewise-smooth all these approaches result in the same notion of area. However, if a surface is very irregular, or rough, then it may not be possible to assign any area at all to it. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the theory 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 the geometric measure theory. A specific example of such an extension is the Minkowski content of a surface.

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. If you consider the math, you'll see the relation between SA and V much more intuitively: V = 4/3 π r³; SA = 4 π r², where r is the radius of the cell. Do the math and the resulting ratio becomes 3/r. If a cell has a radius of 1 μm, the SA:V ratio is 3. Increase the cell's radius to 10 μm and the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Using the previous simple example, we can see how the surface area falls off steeply with increasing volume.

Surface area

Formulas

Surfaces whose area cannot be defined

In chemistry

In biology

See also