The area of Hügelschäffer curves via Taylor series

In this paper, we give new Taylor approximative formulae for the area of the egg-shaped parts of Hügelschäffer curves. Based on a parametrization of the Hügelschäffer curve, a formula for the area of the egg-shaped part of such a curve is derived via elliptic integrals of the first and second kind. Furthermore, new approximative formulae for calculating this area derived from standard and double Taylor approximations are given. A representation of the value $\frac{1}{\pi }$ was also obtained using an appropriate series.