This paper presents exact analytic formulae for the lightcurves of stars experiencing eclipses by planets, considering both quadratic and nonlinear limb darkening. For planets with radii less than one-tenth of the stellar radius, the lightcurve can be approximated by assuming a constant surface brightness over the region blocked by the planet. These formulae are applied to Hubble Space Telescope observations of HD 209458, yielding a planetary-to-stellar radius ratio of 0.1207 ± 0.0003. The paper also provides a simpler and faster method for computing lightcurves using limb-darkening coefficients from model atmospheres, which is useful for detecting, simulating, and parameter fitting planetary transits. The results are validated through comparisons with nonlinear limb-darkening models and practical applications, demonstrating the accuracy and efficiency of the proposed methods.This paper presents exact analytic formulae for the lightcurves of stars experiencing eclipses by planets, considering both quadratic and nonlinear limb darkening. For planets with radii less than one-tenth of the stellar radius, the lightcurve can be approximated by assuming a constant surface brightness over the region blocked by the planet. These formulae are applied to Hubble Space Telescope observations of HD 209458, yielding a planetary-to-stellar radius ratio of 0.1207 ± 0.0003. The paper also provides a simpler and faster method for computing lightcurves using limb-darkening coefficients from model atmospheres, which is useful for detecting, simulating, and parameter fitting planetary transits. The results are validated through comparisons with nonlinear limb-darkening models and practical applications, demonstrating the accuracy and efficiency of the proposed methods.