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Encyclopedia > Surface area

Area is the measure of how much exposed area any two dimensional object has. It is expressed in square units, and is calculated by adding together the areas of all the faces of the object. Area is a physical quantity expressing the size of a part of a surface. ... Area is a physical quantity expressing the size of a part of a surface. ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ...

Note: For 2D figures, the surface area and the area are the same.

Common equations for surface area (2-Dimensional Objects):
Shape Equation Variables
A square: s2 s = length of any side
A rectangle: $l cdot w$ l = length, w = width
A circle: $pi cdot r^2$ r = radius
Any regular polygon: $P cdot a/2$ P = length of the perimeter, a = length of the apothem of the polygon (the distance from the center of the polygon to the center of one side)
A parallelogram: $B cdot h$ B (base) = any side, h (height) = the distance between the lines that the sides of length B lie on
A trapezoid: $(B+b) cdot h/2$ B and b = lengths of the parallel sides, h = distance between the lines on which the parallel sides lie
A triangle (1): $B cdot h/2$ B = any side, h = distance from the line on which B lies to the other point of the triangle
A triangle (2)

(Heron's formula): An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... 2-dimensional renderings (ie. ... For other uses, see Square. ... In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... Circle illustration This article is about the shape and mathematical concept of circle. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... This article or section is not written in the formal tone expected of an encyclopedia article. ... A trapezoid (in North America) or trapezium (in Britain and elsewhere) is a quadrilateral, which is defined as a shape with four sides, which has a pair of parallel sides. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... A triangle with sides a, b, and c. ...

$sqrt{[p cdot (p-a) cdot (p-b) cdot (p-c)]}$ a, b and c = sides of triangle, p = half of the perimeter, or (a+b+c)/2
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
A cylinder: $2 cdot pi cdot r cdot (h+r)$ r = radius of circular base, h = height
A cone: $pi cdot r cdot [(r + sqrt{(r^2+h^2)}]$ 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

## Ill-defined areas

If one adopts the axiom of choice, then it is possible to prove that there are some shapes whose area cannot be meaningfully defined; see Lebesgue measure for more details. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...

Results from FactBites:

 Surface Area Formulas (565 words) In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object. Surface Area of a Cube = 6 a In words, the surface area of a cube is the area of the six squares that cover it.
 Surface Area Lessons by MATHguide (0 words) The definition of surface area is the sum of all unit squares that fit on the exterior of a solid. All the surfaces of a prism are rectangular. The third surface, the lateral surface area, is less easy to visualize for the purposes of calculating its area, especially since it does not appear to be in a shape that fits a known area like a triangle or parallelogram.
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