The heat diffusion equation for an annular fin is analyzed using Laplace transformations. The fin has a uniform thickness with its edge heat loss and two temperature profiles at the base such as a step change in temperature or heat flux. To obtain the exact solutions for temperature distribution, this paper can detect the eigenvalues which satisfy the roots of transcendental equations in above two cases during inverse Laplace transformations. The exact solutions for temperature and heat flux are obtained with the infinite Series by dimensionless factors. The solutions are developed for small and large values of times. These series solutions converge rapidly for large values of time, but slowly for small.