Page "Surface area" Paragraph 6

Surface areas of flat polygonal shapes must agree with their geometrically defined area.

Since surface area is a geometric notion, areas of congruent surfaces must be the same and area must depend only on the shape of the surface, but not on its position and orientation in space.

This means that surface area is invariant under the group of Euclidean motions.

These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth.

Such surfaces consist of finitely many pieces that can be represented in the parametric form