# Area

**Area**is a quantity expressing the size of a region of space.

**Surface area**refers to the summation of the exposed sides of an object. Area (

*Cx*

^{2}) is the derivative of volume (

*Cx*

^{3}). Area is the antiderivative of length (

*Cx*

^{1}).

Table of contents |

2 Some formulas 3 Ill-defined areas 4 External Link |

## Units

Units for measuring surface area include:

- square metre - SI derived unit
- are - 100 square metres
- hectare - 10,000 square metres
- square kilometre - 1,000,000 square metres

- square foot (plural feet) - 0.09290304 square meters.
- square yard - 9 square feet - 0.83612736 square metres
- square perch - 30.25 square yards - 25.2928526 square metres
- acre - 160 square perches or 43,560 square feet - 4046.8564224 square metres
- square mile - 640 acres - 2.5899881103 square kilometres

## Some formulas

For a two dimensional object the area and surface area are the same:

- square or rectangle:
(where l is the length and w is the width; in the case of a square, l = w.*l × w* - circle:
(where r is the radius)*&pi×r*^{2} - any regular polygon:
(where P = the length of the perimeter, and a is the length of the apothem of the polygon [the distance from the center of the polygon to the center of one side])*P × a / 2* - a parallelogram:
(where the base B is any side, and the height h is the distance between the lines that the sides of length B lie on)*B × h* - a trapezoid:
(B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)*(B + b) × h / 2* - a triangle:
(where B is any side, and h is the distance from the line on which B lies to the other point of the triangle). Alternatively,*B × h / 2**Heron's formula*can be used:(where a, b, c are the sides of the triangle, and s = (a + b + c)/2 is half of its perimeter)*√(s×(s-a)×(s-b)×(s-c))* - the area between the graphss of two functions is equal to the integral of one function,
*f*(*x*), minus the integral of the other function,*g*(*x*).

- cube:
, where s is the length of any side*6×(s*^{2}) - rectangular box:
, where l, w, and h are the length, width, and height of the box*2×((l × w) + (l × h) + (w × h))* - sphere:
, where &pi is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere*4×π×(r*^{2}) - cylinder:
, where r is the radius of the circular base, and h is the height*2×π×r×(h + r)* - cone:
, where r is the radius of the circular base, and h is the height.*π×r×(r + √(r*^{2}+ h^{2}))

### See also

*An artist should feel free to add some example diagrams.*

## 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. Such 'shapes' (they cannot *a fortiori* be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach-Tarski paradox). The sets involved will not arise in practical matters.