area
The area of a twodimensional figure is the amount of space contained within the figure. Area is typically measured in ; i.e. (http://planetmath.org/Ie), if the area of a figure is 5 ${\text{in}}^{2}$, this means that, if five 1 inch by 1 inch squares are cut appropriately, they can be arranged so that they exactly cover the space contained in the figure without any overlapping. In formulas^{}, area is almost always denoted using the letter $A$.
All examples provided within this entry are in Euclidean geometry^{}.
For certain figures, area is quite commonly found by multiplying the lengths of two line segments^{} which are related to the figure as well as perpendicular^{} to each other. Below are some examples:

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triangle^{}: $A={\displaystyle \frac{1}{2}}bh$, where $b$ is its base (http://planetmath.org/BaseAndHeightOfTriangle) and $h$ is its height (http://planetmath.org/BaseAndHeightOfTriangle)

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parallelogram^{}: $A=bh$, where $b$ is its base and $h$ is its height

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ellipse^{} (http://planetmath.org/Ellipse2): $A=\pi ab$, where $a$ and $b$ are the major semiaxis and semiaxis (not necessarily respectively)
Also, in a regular^{} $n$gon (http://planetmath.org/RegularPolygon), each apothem is perpendicular to a side of the polygon^{}. Thus, the formula $A={\displaystyle \frac{1}{2}}aP$, where $a$ is the length of its apothem and $P$ is its perimeter^{} (http://planetmath.org/Perimeter2), may be considered as another example.
Any threedimensional figure has a surface which is twodimensional. For certain figures, such as cubes and cylinders, this fact can easily be verified by the surface and it to lie flat. The surface area of a threedimensional figure is the area of its surface.
One method of determining the surface area of any threedimensional figure is by investigating how much paint would be required to cover its surface with exactly one coat of paint. (This works best if the paint is considered to have no thickness.)
There is another method of determining the surface area of a threedimensional figure. It works best on figures whose surfaces can easily be cut and forced to lie flat. Once this is done, the surface area can be obtained by determining the area of the resulting twodimensional figure.
For example, a cube is made up of six congruent^{} (http://planetmath.org/Congruence^{}) squares. If each square has a side of length $s$, then the surface area of the cube is $6{s}^{2}$.
As another example, for a cylinder with radius $r$ and height $h$, its top and bottom, which are circles, can be cut off, and the remaining portion can be unrolled as a rectangle^{}. The radius of each circle is $r$, so they each have area of $\pi {r}^{2}$. The rectangle has a width that is equal to the circumference of the circular faces, and its height is $h$. Thus, the area of the rectangle is $2\pi rh$. Therefore, the surface area of the cylinder is $2\pi {r}^{2}+2\pi rh$.
For some threedimensional figures, determining its surface area in this manner may not be very straightforward. For example, to determine the surface area of a sphere, one could try peeling an orange and making the portions of orange peel lie flat, but it would be very difficult to come up with the correct formula of $4\pi {r}^{2}$ from this procedure. The method of painting as described earlier works much better for spheres.
Remarks

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When the shape of the geometric figure is complicated, the area can be computed using techniques from calculus. The idea is to break up the geometric figure into tiny squares. The area of the figure may be approximated by the total area occupied by these squares. The interesting thing is whether it is possible to get an exact answer if the squares are tiny enough. For all of the examples given above, using the tiny squares will give the exact answer.

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The concept of area is a special case of a general concept called measure, or more appropriately, product measure.
Title  area 
Canonical name  Area 
Date of creation  20130322 16:57:34 
Last modified on  20130322 16:57:34 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  36 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 51M25 
Classification  msc 5100 
Related topic  AreaOfAPolygonalRegion 
Related topic  DevelopableSurface 
Related topic  AreaOfPlaneRegion 
Related topic  Volume2 
Related topic  BasicLength 
Related topic  BaseAndHeightOfTriangle 
Defines  surface area 