Mean radius

The mean radius' (or sometimes the equivalent radius, or volumetric mean radius) in applied sciences is determination of the "average" radius of a non-circular or non-spherical object by treating it as if it were circular or spherical. In astronomy, for instance, the size of planets and small Solar System bodies is often expressed in terms of a mean radius, even though they are not perfect spheres. Alternatively, the closely related mean diameter (${\displaystyle D}$), which is twice the mean radius, is also used.

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

${\displaystyle P=2\pi R_{\text{mean}}}$

or alternatively

${\displaystyle R_{\text{mean}}={\frac {P}{2\pi }}}$

For example, a square of side L has a perimeter of ${\displaystyle 4L}$. Setting that perimeter to be equal to that of a circle imply that

${\displaystyle R_{\text{mean}}={\frac {2L}{\pi }}\approx 0.6366L}$

• 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 1D mean diameter.^[1]
• Diameter at breast height is the circumference of tree trunk, measured at height of 4.5 feet, divided by pi. This corresponds to the 1D mean diameter. It can be measured directly by a girthing tape.^[2]

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 hydraulic diameter is similarly defined as 4 times the cross-sectional area of a pipe A, divided by it's "wetted" perimeter P. For a circular pipe of radius R, at full flow, this is

${\displaystyle D_{\text{H}}={\frac {4\pi R^{2}}{2\pi R}}=2R}$

as one would expect. This is equivalent to the above definition of the 2D mean radius. However, for historical reasons, the hydraulic radius is defined as the cross-sectional area of a pipe A, divided by it's wetted perimeter P, which leads to ${\displaystyle D_{\text{H}}=4R_{\mathbb {H} }}$, and the hydraulic radius is half of the 2D mean radius.^[3]

• In aggregate classification, the equivalent diameter is the "diameter of a circle with an equal aggregate sectional area", which is calculated by ${\displaystyle D=2{\sqrt {\frac {A}{\pi }}}}$. It is used in many digital image processing programs.^[4]

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 ${\displaystyle R_{\text{mean}}={\sqrt[{3}]{a\cdot b\cdot c}}}$.^[5] 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 ${\displaystyle R_{\text{mean}}={\sqrt[{3}]{a^{2}\cdot c}}}$.^[6]

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 3D 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.^[6]
• The asteroid 511 Davida, which is close in shape to a triaxial ellipsoid with dimensions 360 km × 294 km × 254 km, has a 3D mean diameter of ${\displaystyle D={\sqrt[{3}]{360\cdot 294\cdot 254}}=300{\text{ km}}}$.^[7]

• Cubic mean
• Earth ellipsoid
• Geoid
• Geometric mean