Equivalent radius explained
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) (
) is twice the equivalent radius.
Perimeter equivalent
The perimeter of a circle of radius R is
. Given the perimeter of a non-circular object
P, one can calculate its
perimeter-equivalent radius by setting
or, alternatively:
For example, a square of side L has a perimeter of
. Setting that perimeter to be equal to that of a circle imply that
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
The area of a circle of radius R is
. Given the area of a non-circular object
A, one can calculate its
area-equivalent radius by setting
or, alternatively:
}Often the area considered is that of a
cross section.
For example, a square of side length L has an area of
. Setting that area to be equal that of a circle imply that
} L \approx 0.3183 L
and
semi-minor axis
has mean radius
.
For a circle, where
, this simplifies to
.
Applications:
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
, 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
}. It is used in many digital image processing programs.
[4] Volume equivalent
The volume of a sphere of radius R is
. Given the volume of a non-spherical object
V, one can calculate its
volume-equivalent radius by setting
or, alternatively:
}
For example, a cube of side length L has a volume of
. Setting that volume to be equal that of a sphere imply that
} L \approx 0.6204 L
Similarly, a tri-axial ellipsoid with axes
,
and
has mean radius
Rmean=\sqrt[3]{a ⋅ b ⋅ c}
.
[5] The formula for a rotational ellipsoid is the special case where
. Likewise, an oblate spheroid or
rotational ellipsoid with axes
and
has a mean radius of
. For a sphere, where
, this simplifies to
.
Applications:
- For planet Earth, which can be approximated as an oblate spheroid with radii and, the 3D mean radius is
R=\sqrt[3]{6378.12 ⋅ 6356.8}=6371.0km
.
[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
Notes and References
- Book: Bello . Ignacio . Britton . Jack Rolf . 1993 . Topics in Contemporary Mathematics . 5th . 512 . Lexington, Mass . D.C. Heath . 9780669289572.
- Book: West . P. W. . 2004 . Tree and Forest Measurement . Stem diameter . 13ff . New York . Springer . 9783540403906 .
- Wei . Maoxing . Cheng . Nian-Sheng . Lu . Yesheng . October 2023 . Revisiting the concept of hydraulic radius . Journal of Hydrology . 625 . Part B . 130134 . 10.1016/j.jhydrol.2023.130134 . 2023JHyd..62530134W .
- Book: 10.1016/B978-0-12-849908-5.00013-4 . Asphalt mix homogeneity . Structural Behavior of Asphalt Pavements . 2016 . Sun . Lijun . 821–921 . 978-0-12-849908-5 .
- Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination. J.. Leconte. D.. Lai. G.. Chabrier. Astronomy & Astrophysics. 528. A41. 2011. 9. 10.1051/0004-6361/201015811. 1101.2813 . 2011A&A...528A..41L .
- Mean radius, mass, and inertia for reference Earth models. F.. Chambat. B.. Valette. Physics of the Earth and Planetary Interiors. 124. 3–4. 2001. 4. 10.1016/S0031-9201(01)00200-X. 2001PEPI..124..237C .
