Density of the first k coordinates of a uniform random variable

Suppose that X is distributed uniformly in the n-sphere $\sqrt{n}{\mathbf{S}}^{n-1}\subset {\mathbf{R}}^{n}$. Then apparently the distribution of $({X}_{1},\dots ,{X}_{k})$, the first $k<n$ coordinates of X has density $p({x}_{1},\dots ,{x}_{k})$ with respect to Lebesgue measure in ${\mathbf{R}}^{k}$, moreover if ${r}^{2}={x}_{1}^{2}+\cdots +{x}_{k}^{2}$, then it is proportional to

${(1-\frac{{r}^{2}}{n})}^{(n-k)/2-1},\phantom{\rule{1em}{0ex}}\text{if}\text{}0\le {r}^{2}\le n,$

and otherwise is 0. I tried to compute this using the fact that $({X}_{1},\dots ,{X}_{k})\stackrel{\mathrm{d}}{=}\sqrt{n}({g}_{1},\dots ,{g}_{k})/\sqrt{{g}_{1}^{2}+\cdots +{g}_{n}^{2}}$, when ${g}_{i}$ are iid standard normal variables, but was unable to simplify the integrals. Does anyone know/can point me to a place where this density is derived?