We present a fast and accurateapproximation of the Euclidean thicknessdistribution computation of a binary shape in arbitrarydimension. Thickness functions associate a value representing the local thickness for each point of a binary shape. When considering with the Euclidean metric, a simple definition is to associate with each point x, the radius of the largest ball inscribed in the shape containing x. Such thicknessdistributions are widely used in many applications such as medical imaging or material sciences and direct implementations could be time consuming. In this paper, we focus on fast algorithms to extract such distribution on shapes in arbitrarydimension.