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 (Cx2) is the derivative of volume (Cx3). Area is the antiderivative of length (Cx1).

Table of contents
1 Units
2 Some formulas
3 Ill-defined areas
4 External Link


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

Old British units, as currently defined from the metre:
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

The article Orders of magnitude links to lists of objects of comparable surface area.

Some formulas

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

  • square or rectangle: l w (where l is the length and w is the width; in the case of a square, l = w.
  • circle: &pi×r2 (where r is the radius)
  • any regular polygon: P a / 2 (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])
  • a parallelogram: B h (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)
  • a trapezoid: (B + b) h / 2 (B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)
  • a triangle: B h / 2 (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, Heron's formula can be used: √(s×(s-a)×(s-b)×(s-c)) (where a, b, c are the sides of the triangle, and s = (a + b + c)/2 is half of its perimeter)
  • 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).

Some basic formulas for calculating surface areas of three dimensional objects are:

  • cube: 6(s2) , where s is the length of any side
  • rectangular box: 2((l w) + (l h) + (w h)), where l, w, and h are the length, width, and height of the box
  • sphere: 4π(r2) , where &pi is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere
  • cylinder: 2×π×r×(h + r), where r is the radius of the circular base, and h is the height
  • cone: π×r×(r + √(r2 + h2)), where r is the radius of the circular base, and h is the height.

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.

External Link