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 (), which is twice the mean radius, is also used.
1D
The perimeter of a circle of radius R is . Given the perimeter of a non-circular object P, one can calculate its mean 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
Examples
- 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]
2D
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Wetted_Perimeter.svg/220px-Wetted_Perimeter.svg.png)
The area of a circle of radius R is . Given the area of an non-circular object A, one can calculate its mean radius by setting
or alternatively
For example, a square of side length L has an area of . Setting that area to be equal that of a circle imply that
Similarly, an ellipse with semi-major axis and semi-minor axis has mean radius .
For a circle, where , this simplifies to .
Example
- 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
- 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 , 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]
3D
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/300px-Ellipsoide.svg.png)
The volume of a sphere of radius R is . Given the volume of an non-spherical object V, one can calculate its mean 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
Similarly, a tri-axial ellipsoid with axes , and has mean radius .[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 .[6]
For a sphere, where , this simplifies to .
Examples
- 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 . The equatorial and polar radii of a planet are often denoted and , 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 .[7]
See also
References
- ^ Bello, Ignacio; Britton, Jack Rolf (1993). Topics in Contemporary Mathematics (5th ed.). Lexington, Mass: D.C. Heath. p. 512. ISBN 9780669289572.
- ^ West, P. W. (2004). "Stem diameter". Tree and Forest Measurement. New York: Springer. pp. 13ff. ISBN 9783540403906.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ a b 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.
- ^ Ridpath, I. (2012). "Davida". A Dictionary of Astronomy. Oxford University Press. p. 115. ISBN 978-0-19-960905-5.