Equivalent radius

Area equivalent

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

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

or, alternatively:

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

Often the area considered is that of a cross section.

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

${\displaystyle R_{\text{eq}}=\sqrt{\frac{L^2}{\pi }}=\sqrt{\frac{1}{\pi }}L\approx 0.5642L}$

Similarly, an ellipse with semi-major axis ${\displaystyle a}$ and semi-minor axis ${\displaystyle b}$ has area of ${\displaystyle \pi ab}$, and therefore

${\displaystyle R_{\text{eq}}=\sqrt{\frac{\pi ab}{\pi }}=\sqrt{ab}}$.

Applications:

• The hydraulic diameter is similarly defined as 4 times the cross-sectional area of a pipe A, divided by its "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 diameter. However, for historical reasons, the hydraulic radius is defined as the cross-sectional area of a pipe A, divided by its wetted perimeter P, which leads to ${\displaystyle D_{\text{H}}=4R_{\text{H}}}$, and the hydraulic radius is half of the 2D mean radius.

• 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.

Volume equivalent

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

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

or, alternatively:

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

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

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

Similarly, a tri-axial ellipsoid with axes ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ has a volume of ${\displaystyle \frac{4}{3}\pi abc}$, and therefore

${\displaystyle R_{\text{eq}}=\sqrt[3]{\frac{3\frac{4}{3}\pi abc}{4\pi }}=\sqrt[3]{abc}}$.

The formula for a rotational ellipsoid is the special case where ${\displaystyle a=b}$

${\displaystyle R_{\text{eq}}=\sqrt[3]{a^2\cdot c}}$.

Applications:

• 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 ${\displaystyle R=\sqrt[3]{{6378.1}^2\cdot 6356.8}=6371.0\text{ km}}$.

Other equivalences

Surface-area equivalent radius

The surface area of a sphere of radius R is ${\displaystyle 4\pi R^2}$. Given the surface area of a non-spherical object A, one can calculate its surface area-equivalent radius by setting

${\displaystyle 4\pi R_{\text{eq}}^2=A}$

or equivalently

${\displaystyle R_{\text{eq}}=\sqrt{\frac{A}{4\pi }}}$

For example, a cube of length L has a surface area of ${\displaystyle 6L^2}$. A cube therefore has an surface area-equivalent radius of

${\displaystyle R_{\text{eq}}=\sqrt{\frac{6L^2}{4\pi }}\approx 0.6910L}$

Curvature-equivalent radius

The osculating circle and osculating sphere define curvature-equivalent radii at a particular point of tangency for plane figures and solid figures, respectively.