|Definition:|| unique positive value, if it exists, associated with a subset of a surface in the three-dimensional Euclidean space, with the following properties: |
- 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 to entry: 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 .
Note 2 to entry: For a surface defined by where , the area is .
Note 3 to entry: For a surface defined by the equation z = f(x, y), the area is .
Note 4 to entry: In the usual geometrical space, the area of a surface is a quantity of the dimension length squared.