Mean radius

The mean radius (or sometimes the volumetric mean radius) in astronomy is a measure for the size of planets and small Solar System bodies. Alternatively, the closely related mean diameter ($$D$$), which is twice the mean radius, is also used. For a non-spherical object, the mean radius (denoted $$R$$ or $$r$$) is defined as the radius of the sphere that would enclose the same volume as the object. In the case of a sphere, the mean radius is equal to the radius.

For any irregularly shaped rigid body, there is a unique ellipsoid with the same volume and moments of inertia. The dimensions of the object are the principal axes of that special ellipsoid.

Calculation
The volume of a sphere of radius R is $$\frac{4}{3}\pi R^3$$. Given the volume of an non-spherical object V, one can calculate its mean radius by setting
 * $$V = \frac{4}{3}\pi R^3_\text{mean}$$

or alternatively
 * $$R_\text{mean} = \sqrt[3]{\frac{3V}{4\pi}}$$

For example, a cube of side length L has a volume of $$L^3$$. Setting that volume to be equal that of a sphere imply that
 * $$R_\text{mean} = \sqrt[3]{\frac{3}{4\pi}} L \approx 0.6204 L$$

Similarly, a tri-axial ellipsoid with axes $$a$$, $$b$$ and $$c$$ has mean radius $$R_\text{mean}=\sqrt[3]{a \cdot b \cdot c}$$. The formula for a rotational ellipsoid is the special case where $$a=b$$.

Likewise, an oblate spheroid or rotational ellipsoid with axes $$a$$ and $$c$$ has a mean radius of $$R_\text{mean}=\sqrt[3]{a^{2} \cdot c }$$.

For a sphere, where $$a=b=c$$, this simplifies to $$R_\text{mean}=a$$.

Examples

 * For planet Earth, which can be approximated as an oblate spheroid with radii $6,378.1 km$ and $6,356.8 km$, the mean radius is $$R=\sqrt[3]{6378.1^{2}\cdot6356.8}=6371.0\text{ km}$$. The equatorial and polar radii of a planet are often denoted $$r_{e}$$ and $$r_{p}$$, respectively.
 * The asteroid 511 Davida, which is close in shape to a triaxial ellipsoid with dimensions $360 km$, has a mean diameter of $$D=\sqrt[3]{360\cdot294\cdot254}=300\text{ km}$$.