We are interested in finite cones of fixed height 1 parametrized by their opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when their apertures tend to 0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues of each fibers of the Dirichlet Laplacian. To do so, we investigate the family of their one-dimensional Born-Oppenheimer approximations. The eigenvalue asymptotics involves powers of the cube root of the aperture, while the eigenvectors include simultaneously two scales.