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 × 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:
4×π×(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.
