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). In the case of the perfect closed curve in two dimensions, which is the circle, the area is the simple integral of the circumference. Thus, the circumference is 2πr, while the area is πr².

Table of contents

1 Units
2 Some formulas

2.1 See also

3 How to define area
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 megametre - 10¹² 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 ÃÂÃÂ 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 ÃÂÃÂ 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 ÃÂÃÂ 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) ÃÂÃÂ 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 ÃÂÃÂ h / 2 (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then 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: 6ÃÂÃÂ(s²) , where s is the length of any side
rectangular box: 2ÃÂÃÂ((l ÃÂÃÂ w) + (l ÃÂÃÂ h) + (w ÃÂÃÂ h)), where l, w, and h are the length, width, and height of the box
sphere: 4ÃÂÃÂπÃÂÃÂ(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.

How to define area

Although area seems to be one of the basic notions in geometry it is not at all easy to define, even in Euclidean plane. Most books avoid it wrongly referring to self-evidence. To make area meaningful one has at least define it on polygones in Euclidean plane, and it can be done using following definition:

Area of polygon in Eucledian plane is a positive number such that
Area of unit square is equal one.
Equal polygons have equal area.
If a polygon is a union of two polygons which do not have coommon interior points then its area is sum of areas of these polygons.

But before using this definition one has to prove that such an area indeed exist.

One can also give a formula for area of triangle, and then define the area of polygon using that area of union of polygons (without coommon interior points) is sum of areas of its pieces. But then it is not quite easy to show that such area does not depends on the way you break the polygon into pieces.

Nowadays most standard (correct) way to introduce area is through more advanced notion of Lebesgue measure, but one should note that in general, if one adopts the axiom of choice then it is possible to prove that there are some shapes whose Lebegue measure cannot be meaningfully defined. 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 do not arise in practical matters.

External Link

Conversion Calculator for Units of AREA