The distance between the origin and the centroid of the tetrahedron whose vertices are $\left(0,0,0\right),\left(\mathrm{a},0,0\right),\left(0,\mathrm{b},0\right),\left(0,0,\mathrm{c}\right)$ is 

If $G$ is centroid of the tetrahedron $ABCD$ then the $G=\frac{A+B+C+D}{4}$

The centroid of the tetrahedron is $G\left(\frac{a}{4},\frac{b}{4},\frac{c}{4}\right)$

The distance from the centroid of the tetrahedron and origin is $\sqrt{\frac{a^2}{16}+\frac{b^2}{16}+\frac{c^2}{16}}=\frac{1}{4}\sqrt{a^2+b^2+c^2}$