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 (${\displaystyle D}$), which is twice the mean radius, is also used. For a non-spherical object, the mean radius (denoted ${\displaystyle R}$ or ${\displaystyle r}$) is defined as the radius of the sphere that would enclose the same volume as the object.^[1] 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.^[2] The dimensions of the object are the principal axes of that special ellipsoid.^[3]

The area of a circle of radius R is ${\displaystyle \pi R^{2}}$. Given the area of an non-circular object A, one can calculate its mean radius by setting

${\displaystyle A=\pi R_{\text{mean}}^{2}}$

or alternatively

${\displaystyle R_{\text{mean}}={\sqrt {\frac {A}{\pi }}}}$

For example, a square of side length L has an area of ${\displaystyle L^{2}}$. Setting that area to be equal that of a circle imply that

${\displaystyle R_{\text{mean}}={\sqrt {\frac {1}{\pi }}}L\approx 0.3183L}$

Similarly, an ellipse with semi-major axis ${\displaystyle a}$ and semi-minor axis ${\displaystyle b}$ has mean radius ${\displaystyle R_{\text{mean}}={\sqrt {a\cdot b}}}$.

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

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

${\displaystyle V={\frac {4}{3}}\pi R_{\text{mean}}^{3}}$

or alternatively

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

For example, a cube of side length L has a volume of ${\displaystyle L^{3}}$. Setting that volume to be equal that of a sphere imply that

${\displaystyle R_{\text{mean}}={\sqrt[{3}]{\frac {3}{4\pi }}}L\approx 0.6204L}$

Similarly, a tri-axial ellipsoid with axes ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ has mean radius ${\displaystyle R_{\text{mean}}={\sqrt[{3}]{a\cdot b\cdot c}}}$.^[1] The formula for a rotational ellipsoid is the special case where ${\displaystyle a=b}$.

Likewise, an oblate spheroid or rotational ellipsoid with axes ${\displaystyle a}$ and ${\displaystyle c}$ has a mean radius of ${\displaystyle R_{\text{mean}}={\sqrt[{3}]{a^{2}\cdot c}}}$.^[4]

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

• For planet Earth, which can be approximated as an oblate spheroid with radii 6378.1 km and 6356.8 km, the mean radius is ${\displaystyle R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}}$. The equatorial and polar radii of a planet are often denoted ${\displaystyle r_{e}}$ and ${\displaystyle r_{p}}$, respectively.^[4]
• The asteroid 511 Davida, which is close in shape to a triaxial ellipsoid with dimensions 360 km × 294 km × 254 km, has a mean diameter of ${\displaystyle D={\sqrt[{3}]{360\cdot 294\cdot 254}}=300{\text{ km}}}$.^[5],
• US hat size is the circumference of the head, measured in inches, divided by pi, rounded to the nearest 1/8 inch. This corresponds to the mean diameter.^[6]
• Diameter at breast height is the circumference of tree trunk, measured at height of 4.5 feet, divided by pi. This corresponds to the mean diameter. It can be measured directly by a girthing tape.^[7]

• Cubic mean
• Earth ellipsoid
• Geoid
• Geometric mean