A signed area, also called an oriented area or algebraic area, assigns a sign to the area enclosed by an oriented closed
curve. For a piecewise smooth parametrized curve 
,
the signed area is the line integral
 |
(1)
|
or equivalently
![A_s(C)=1/2int_(t_0)^(t_1)[x(t)y^'(t)-y(t)x^'(t)]dt.](/images/equations/SignedArea/NumberedEquation2.svg) |
(2)
|
Reversing the curve orientation reverses the sign,
 |
(3)
|
For a triangle traversed in the order 
, 
, 
, take the left side of the line directed from 
to 
as positive. The signed area is 
, where 
is the length of 
and 
is the signed distance
of 
from that line.
For an oriented polar curve 
,
the corresponding formula is
 |
(4)
|
Equivalently, signed area is the winding-number integral
 |
(5)
|
where 
is the contour winding number of 
about 
. Thus for a self-intersecting
curve, signed area differs in general from both the sum of the absolute areas of
the lobes and the area of their geometric union. This distinction also records repeated
traversals, even when two parametrized curves have the same image.
