• for a rectangle, the value is the product of the two side lengths,
• for a disjoint union of subsets, the value is the sum of the values associated with these subsets,
• for more complicated subsets, the value can be approximated by sums and given by an integral

NOTE 1 For the portion of plane limited by the straight lines x = a, x = b, y = 0 and the arc of curve y = f(x) with a < b and f(x) ≥ 0, the area is ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)\mathrm{d}x}$.

NOTE 2 For a surface defined by $r=f(u,v)$ where $(u,v)\in \text{U}\subseteq R^{\text{2}}$, the area is ${\displaystyle \underset{\text{U}}{\iint }\left|\frac{\partial f}{\partial u}\cdot \frac{\partial f}{\partial u}\right|\cdot \mathrm{d}u\mathrm{d}v}$.

NOTE 3 For a surface defined by the equation z = f(x, y), the area is ${\displaystyle \underset{\text{S}}{\iint }\sqrt{1+{\left(\frac{\partial f}{\partial x}\right)}^2+{\left(\frac{\partial f}{\partial y}\right)}^2}\mathrm{d}x\mathrm{d}y}$.

NOTE 4 In the usual geometrical space, the area of a surface is a quantity of the dimension length squared.