Equivalent radius

Radius of a circle or sphere equivalent to a non-circular or non-spherical object

In applied sciences, the equivalent radius (or mean radius) is the radius of a circle or sphere with the same perimeter, area, or volume of a non-circular or non-spherical object. The equivalent diameter (or mean diameter) ( ${\displaystyle D}$) is twice the equivalent radius.

Perimeter equivalent

Measurement of tree circumference, the tape calibrated to show diameter, at breast height. The tape assumes a circular shape.

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 perimeter-equivalent 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

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

Applications:

• 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]

Area equivalent

Cross sectional area of a trapezoidal open channel, red highlights the wetted perimeter, where water is in contact with the channel. The hydraulic diameter is the equivalent circular configuration with the same circumference as the wetted perimeter.

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 area-equivalent radius by setting

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

or, alternatively:

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

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

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

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 ${\displaystyle R_{\text{mean}}=a}$.

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 ${\displaystyle D_{\text{H}}=4R_{\mathbb {H} }}$, 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]

Volume equivalent

A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right)

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 volume-equivalent 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 ${\displaystyle R_{\text{mean}}=a}$.

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 ${\displaystyle R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}}$.^[6]

Other equivalences

The authalic radius is an surface area-equivalent radius for solid figures such as an ellipsoid.

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

See also

• Antenna equivalent radius
• Cloud drop effective radius
• Cubic mean
• Earth ellipsoid
• Earth radius
• Galaxy effective radius
• Geoid
• Geometric mean
• Semidiameter

References

1. Bello, Ignacio; Britton, Jack Rolf (1993). Topics in Contemporary Mathematics (5th ed.). Lexington, Mass: D.C. Heath. p. 512. ISBN 9780669289572.
2. West, P. W. (2004). "Stem diameter". Tree and Forest Measurement. New York: Springer. pp. 13ff. ISBN 9783540403906.
3. Wei, Maoxing; Cheng, Nian-Sheng; Lu, Yesheng (October 2023). "Revisiting the concept of hydraulic radius". Journal of Hydrology. 625 (Part B): 130134. Bibcode:2023JHyd..62530134W. doi:10.1016/j.jhydrol.2023.130134.
4. Sun, Lijun (2016). "Asphalt mix homogeneity". Structural Behavior of Asphalt Pavements. pp. 821–921. doi:10.1016/B978-0-12-849908-5.00013-4. ISBN 978-0-12-849908-5.
5. Leconte, J.; Lai, D.; Chabrier, G. (2011). "Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination" (PDF). Astronomy & Astrophysics. 528 (A41): 9. arXiv:1101.2813. Bibcode:2011A&A...528A..41L. doi:10.1051/0004-6361/201015811.
6. Chambat, F.; Valette, B. (2001). "Mean radius, mass, and inertia for reference Earth models" (PDF). Physics of the Earth and Planetary Interiors. 124 (3–4): 4. Bibcode:2001PEPI..124..237C. doi:10.1016/S0031-9201(01)00200-X.