Abstract

This paper presents an elementary derivation of the oriented area formula Area(D)= {1} over {2} int _{a} ^{b} {(x(t)y prime (t)−y(t)x prime (t))} dt for a parametrized simple closed curve. Instead of taking Green’s theorem as a starting point, we construct inscribed polygons from a partition of the parameter interval, apply the Shoelace formula to each polygon, and pass to the limit through a Riemann-sum argument. This approach makes the geometric origin of the area integrand explicit and provides a bridge between polygonal area computation and line-integral area formulas. As an application, we compare two similar four-petal-type curves and verify that they have the same enclosed area while having slightly different perimeters.