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

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

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 x w (where l is the length and w is the width;
     in the case of a square, l = w.
   * circle: π×r2 (where r is the radius)
   * any regular polygon: P x 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 x 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) x 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 x 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 graphs 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: 6x(s2) , where s is the length of any side
   * rectangular box: 2x((l x w) + (l x h) + (w x h)), where l, w, and h are
     the length, width, and height of the box
   * sphere: 4xπx(r2) , where π 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.

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.

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