Hausdorff Dimension of a k-dimensional submanifold, hence $L^n(M) = 0$

Question

If $M \in P(R^n)$ is a k-dimensional submanifold, show that the Hausdorff-Dimension of M is k.

Also, is there a quicker/easier way to show, that $L^n(M) = 0$, where L is the Lebesgue-Measurement.