dimension of affine cone

Question

Suppose $X$ is a projective scheme over a field $k$, i.e. $Proj(S)$ for some finitely generated $k$-algebra $S$. Let $C(X)=Spec(S)$ be the affine cone over $X$. Is it always true that $\dim C(X)=\dim X+1$? If not, perhaps there are suitable hypotheses on $X$ (e.g. $X$ is a variety) under which this is true?