Do all compact manifolds have finite volume?

Question

Let $M$ be a compact $k$-dimensional manifold embedded in $\mathbb{R}^d$. Does $M$ being compact imply that the $k$-dimensional volume of $\operatorname{vol}_{k}(M)$ is finite?

Answer 1

Yes: any compact manifold $M$ has finite volume with respect to any Riemannian metric. Given any point $p\in M$, we can find a bounded coordinate chart around $p$ on which each component of the metric is bounded, so that coordinate chart will have finite volume. Since $M$ is compact, it is covered by finitely many of these finite-volume coordinate charts, and so $M$ has finite volume.