The influence of thermal fluctuations on the adhesion behavior of fluid vesicles is investigated with the help of Monte Carlo simulations. The adhesion area A(ad) of a fluid vesicle adhering to a smooth attractive substrate is studied systematically for different values of temperature, adhesion strength, and potential range. For low temperatures T , the ratio A(ad) /A between the adhesion area and the total area A of the vesicle is a linear function of T/kappa , where kappa is the bending rigidity. Linear fits of the simulation data allow an extrapolation to T=0 which corresponds well with data obtained from a simplified analytic model. A new ansatz for A(ad) (T) which is based on the eigenmodes of the adhering vesicle explains the linear behavior of A(ad) (T) for low T and helps to define a fit function which reproduces the linear behavior of the obtained simulation data. This fit function may be used in order to determine the bending rigidity and the adhesion strength from the observed adhesion geometry.