For other uses see Area (disambiguation).
Area is a quantity expressing the twodimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of surfaces.
Units for measuring area include:
are (a) = 100 square meters (m²)
hectare (ha) = 100 ares (a) = 10000 square meters (m²)
square kilometre (km²) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (m²)
square megametre (Mm²) = 10^{12} square metres
square foot = 144 square inches = 0.09290304 square metres (m²)
square yard = 9square feet = 0.83612736 square metres (m²)
square perch = 30.25 square yards = 25.2928526 square metres (m²)
acre = 10 square chains (also one furlong by one chain); or 160 square perches; or 4840 square yards; or 43560square feet = 4046.8564224 square metres (m²)
square mile = 640acres = 2.5899881103 square kilometers (km²)
Square  s^{2}  s  
Regular triangle (equilateral triangle) 
 s  
Regular hexagon 
 s  
Regular octagon  2\left(1+\sqrt{2}\right)s^{2}  s  
Any regular polygon 
ap  a p  
Any regular polygon 
 s n  
Any regular polygon (using degree measure) 
 s n  
Rectangle  lw  l w  
Parallelogram (in general)  bh  b h  
Rhombus 
ab  a b  
Triangle 
bh  b h  
Triangle 
ab\sin(C)  a b C  
Circle  \pi
 r d  
Ellipse  \piab  a b  
Trapezoid 
(a+b)h  a b h  
Total surface area of a Cylinder  2\pir^{2+2\pi}rh  r h  
Lateral surface area of a cylinder  2\pirh  r h  
Total surface area of a Cone  \pir(l+r)  r l  
Lateral surface area of a cone  \pirl  r l  
Total surface area of a Sphere  4\pir^{2 or \pi}d^{2}  r d  
Total surface area of an ellipsoid  See the article.  
Circular sector 
r^{2}\theta  r \theta  
Square to circular area conversion 
A  A  
Circular to square area conversion 
C\pi  C  
A revolution of f(x) about the xaxis  2\pi
 f(x)  \sqrt  
Area of surface of revolution of f(x) about the yaxis  2\pi
 x  \sqrt 
All of the above calculations show how to find the area of many shapes.
The area of irregular polygons can be calculated using the "Surveyor's formula".^{[1]}
Area is a quantity expressing the size of the contents of a region on a 2dimensional surface. Points and lines have zero area, cf. spacefilling curves. A region may have infinite area, for example the entire Euclidean plane. The 3dimensional analog of area is volume. Although area seems to be one of the basic notions in geometry, it is not easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on selfevidence. For polygons in the Euclidean plane, one can proceed as follows:
The area of a polygon in the Euclidean plane is a positive number such that:
It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts.
A typical way to introduce area is through the more advanced notion of Lebesgue measure. In the presence of the axiom of choice it is possible to prove the existence of shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circlesquaring problem (and, moving to three dimensions, in the Banach–Tarski paradox). The sets involved do not arise in practical matters.
In three dimensions, the analog of area is called volume. The n dimensional analog, usually referred to as 'content', is defined by means of a measure or as a Lebesgue integral.
Bh  
2 
\sqrt{s(sa)(sb)(sc)}
s=
a+b+c  
2 
{1\over2}
2\pi  
\int  
0 
r^{2}d\theta
\vecu(t)=(x(t),y(t))
\vecu(t_{0)}=\vecu(t_{1)}
t_{1}  
\oint  
t_{0} 
x
y 
dt=
t_{1}  
\oint  
t_{0} 
y
x 
dt={1\over2}
t_{1}  
\oint  
t_{0} 
(x
y 
y
x) 
dt
or the zcomponent of
{1\over2}
t_{1}  
\oint  
t_{0} 
\vecu x
\vecu 
dt.
6s^{2}
2(\ellw+\ellh+wh)
\pir\left(r+\sqrt{r^{2}+h^{2}\right)}
\pir^{2}+\pirl
\pir^{2}
\pirl
2 * Area of Base + Perimeter of Base * Height
The general formula for the surface area of the graph of a continuously differentiable function
z=f(x,y),
(x,y)\inD\subsetR^{2}
D
A=\iint  

 
\right) 
\right)^{2+1}dxdy.}
r=r(u,v),
r
(u,v)\inD\subsetR^{2}
A=\iint_{D}\left
\partialr  x  
\partialu 
\partialr  
\partialv 
\rightdudv.
Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.
The question of the filling area of the Riemannian circle remains open.