First, let's consider a rectangle. On a coordinate plane, we'll say it has vertices at (0,0), (a,0), (0,b), and (a,b) Then, an orthogonal parametrization of the boundary can be obtained by taking F(x,y) = (ax, by) The domain of F is D = {(x,y): 0 <= x <= 1, 0 <= y <= 1} Our general formula claims that: Area = integral (sqrt [[ Fx*Fx, Fx*Fy][Fy*Fx, Fy*Fy]] over the domain D. The * operation in this case is the vector scalar product. Since the matrix is positive definite, the root of its determinant will never be imaginary so we will always have a positive or zero real area. The partial derivatives can be easily computed to be Fx = (a, 0) and Fy = (0, b) Evaluating the determinant yields a^2*b^2, so that sqrt(a^2*b^2) = a*b Integrating over [0,1] with respect to x and over [0,1] with respect to y yields Area = (a*b)*1*1 = a*b So we see that the area of a rectangle is, perhaps not obviously, the product of the lengths of its sides. [Edited on 12.10.2010 12:21 PM PST]