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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.

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.

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: 4×π×(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.

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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

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