We prove a priori anisotropic estimates for the $L^2$ and $H^1$ interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new $H^1$ error estimate does not require the “maximal angle condition”. The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results.