Surface area

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

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 Imperial 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.

Surface area formulae

Note: For 2D figures, the surface area and the area are the same.

Common equations for surface area (2-Dimensional Objects):
Shape	Equation	Variables
A square:	$s^{2}$	s = length of any side
A rectangle:	$l\cdot w$	l = length, w = width
A circle:	$\pi \cdot r^{2}$	r = radius
Any regular polygon:	$P\cdot a/2$	P = length of the perimeter, a = length of the apothem of the polygon (the distance from the center of the polygon to the center of one side)
A parallelogram:	$B\cdot h$	B (base) = any side, h (height) = the distance between the lines that the sides of length B lie on
A trapezoid:	$(B+b)\cdot h/2$	B and b = lengths of the parallel sides, h = distance between the lines on which the parallel sides lie
A triangle (1):	$B\cdot h/2$	B = any side, h = distance from the line on which B lies to the other point of the triangle
A triangle (2) (Heron's formula):	${\sqrt {[p\cdot (p-a)\cdot (p-b)\cdot (p-c)]}}$	a, b and 'c = sides of triangle, p = half of the perimeter, or (a+b+c)/2

Common equations for surface area (3-Dimensional Objects):
Shape	Equation	Variables
A cube:	$6\cdot s^{2}$	s = length of any side
A rectangular prism:	$2\cdot (l\cdot w+l\cdot h+w\cdot h)$	l = length, w = width, h = "h"eight
A sphere:	$4\cdot \pi \cdot r^{2}$	r = radius of sphere
A cylinder:	$2\cdot \pi \cdot r\cdot (h+r)$	r = radius of circular base, h = height
A cone:	$\pi \cdot r\cdot [(r+{\sqrt {(r^{2}+h^{2})}}]$	r = radius of circular base, "h" = 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 for more details.

See also

• Synthetic geometry