The optical Radius of Gyration $$R_g$$ is defined as:

\begin{gather*} R_g^2 = \int r^2 g(r)dr \end{gather*} Where $$r$$ is the distance from a reference point and $$g(r)$$ is the so-called pair distance function:

\begin{gather} g(r) = r^2 \int \Delta \rho(\mathbf{r}')\Delta \rho(\mathbf{r} - \mathbf{r}') d^3 \mathbf{r}'\end{gather}

where $$\Delta \rho(r)$$ is the scattering length density. It's worth noting that $$g(r)$$ contains information about shape and size of the particle and/or macromolecule, to each particle shape corresponds a  well defined $$g(r)$$.

The radius of gyration can be obtained from static light scattering by performing either a Guinier or Zimm Plot.

## Relation of the Radius of Gyration to the geometry of homogeneous objects:

Sphere

For a sphere the radius of gyration is related to the spheres geometric radius, $$R$$ in the following way:\begin{gather} R^2_g = \frac{3}{5} R^2\end{gather}.

Spherical shell

Spherical shell with outer and inner radii $$R_{\mathrm{o}}$$ and $$R_{\mathrm{i}}$$, respectively:\begin{gather} R^2_g = \frac{3}{5} \frac{ R^5_{\mathrm{o}} - R^5_{\mathrm{i}} }{ R^3_{\mathrm{o}} - R^3_{\mathrm{i}} } \end{gather} .

Cylinder

Cylinder with radius $$R$$ and length $$h$$:\begin{gather}R^2_g = \frac{ R^2}{2} + \frac{h^2}{12} \end{gather} .

Hollow Cylinder

Hollow cylinder of length $$h$$ with outer and inner radii $$R_{\mathrm{o}}$$ and $$R_{\mathrm{i}}$$, respectively:\begin{gather}R^2_g = \frac{ R_{\mathrm{o}}^2 + R_{\mathrm{i}}^2}{2} + \frac{h^2}{12} \end{gather} .

Ellipsoid

Ellipsoid with semiaxes a,b, and c:\begin{gather}R^2_g = \frac{ a^2 + b^2 + c^2}{5}  \end{gather} .

Disk

Flat disk with radius $$R$$:\begin{gather}R^2_g = \frac{ R^2 }{2}  \end{gather} .

It is advisable to read the different sections in the suggested order if you want to understand all the details. For a quick reference you can also jump to each section individually.

A very detailed source of information is the slide show "Light Scattering Fundamentals".