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Surface area

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Surface area is the measure of how much exposed area any two- or three-dimensional object has.

Table of contents


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: π×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)

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 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.

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 for more details.

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