Surface area is the measure of how much exposed area an object has. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. 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. In Geometry, a face of a Polyhedron is any of the Polygons that make up its boundaries Even objects with smooth surfaces, such as spheres, have surface area.

## Formulas

Sphere: The surface area of a sphere is the integral of infinitesimal circular rings of width dx
The radius of the circular ring is $f(x) = \sqrt{r^2-x^2}$. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The length of the circular ring is equal to $2\pi\cdot f(x)$
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 $2\pi f(x)\cdot \sqrt{1+f'(x)^2}\,dx$
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

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

The antiderivative needed is the simple linear function x
Thus, the sphere surface area amounts to

Asphere = $2\pi r[r-(-r)] = 4\pi r^2 \ \!$

## 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. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative Various approaches to defining the surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish 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. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" 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. In Mathematics, geometric measure theory ( GMT) is the study of the geometric properties of the measures of sets (typically in A specific example of such an extension is the Minkowski content of a surface. The Minkowski content of a set or the boundary measure, is a basic concept in Geometry and Measure theory which generalizes to arbitrary

## In chemistry

Surface area is important in chemical kinetics. Chemical kinetics, also known as reaction kinetics is the study of rates of chemical processes Increasing the surface area of a substance generally increases the rate of a chemical reaction. The reaction rate or rate of reaction for a Reactant or product in a particular reaction is intuitively defined as how fast a reaction takes A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called For example, iron in a fine powder will combust, while in solid blocks it is stable enough to use in structures. Iron (ˈаɪɚn is a Chemical element with the symbol Fe (ferrum and Atomic number 26 Combustion or burning is a complex sequence of Exothermic chemical reactions between a Fuel and an Oxidant accompanied by the production of For different applications a minimal or maximal surface area may be desired.

## In biology

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. The cell is the structural and functional unit of all known living Organisms It is the smallest unit of an organism that is classified as living and is often called The cell membrane (also called the plasma membrane, plasmalemma, or "phospholipid bilayer" is a Selectively permeable Lipid bilayer 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.