  In this paper, we give new Taylor approximative formulae for the area of the
egg-shaped parts of H\"ugelsch\"affer curves. Based on a parametrization of the
H\"ugelsch\"affer 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.
