In this work we provide a novel approach to homogenization for a class of static Hamilton–Jacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach allows us to conclude that the homogenized equation also induces a metric. The advantage of the method is that we can solve just one auxiliary equation to recover the homogenized Hamiltonian $\bar{H}(p)$. This is a significant improvement over existing methods which require the solution of the cell problem (or a variational problem) for each value of p. Computational results are presented and compared with analytic results when available for piecewise constant periodic and random speed functions.