= Radius of gyration of a uniform sphere
{title2=$R_g^2=3R^2/5$}
For a uniform solid <sphere> of radius $R$, the squared <radius of gyration> about its centre is the mass-weighted mean of $r^2$:
$$
R_g^2=\frac{\int_0^Rr^2\,4\pi\rho r^2dr}{4\pi\rho R^3/3}=\frac35R^2.
$$
Thus its volume is $(4\pi/3)(5/3)^{3/2}R_g^3$, about $9.01R_g^3$. A nonuniform object or a different shape need not obey this conversion. This distinction matters when estimating molecular <number density> from scattering measurements of a <radius of gyration>.
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