You can use surface integrals to calculate areas. First, you obtain analytical expressions of the shape’s boundary and then evaluate the double integrals using these expressions. Indeed, this is an advancement in finding areas. However, finding analytical expressions for complex shapes is a challenge. The nineteenth-century mathematician George Green (1793–1841) introduced a remarkable theorem that converted surface integrals to line integrals. An application of this theorem is calculating planar areas using the periphery. All you need to do is specify the coordinates of sufficiently many coordinates on the boundary. Then, the line integral finds shape’s area, be it a pentagram, a concave polygon, or a complex shape like the island of Cyprus.