Equivalent radius

Image:Using a DTApe.JPG

The perimeter of a circle of radius R is $2\; \backslash pi\; R$. Given the perimeter of a non-circular object P, one can calculate its perimeter-equivalent radius by setting

:$P\; =\; 2\backslash pi\; R\_\backslash text\{eq\}$

or, alternatively:

:$R\_\backslash text\{eq\}\; =\; \backslash frac\{P\}\{2\backslash pi\}$

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

:$R\_\backslash text\{eq\}\; =\; \backslash frac\{2L\}\{\backslash pi\}\; \backslash approx\; 0.6366\; L$

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.{{cite book |last1=Bello |first1=Ignacio |last2=Britton |first2=Jack Rolf |date=1993 |title=Topics in Contemporary Mathematics |edition=5th |page=512 |location=Lexington, Mass |publisher=D.C. Heath |isbn=9780669289572}}
• 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.{{cite book |last1=West |first1=P. W. |date=2004 |title=Tree and Forest Measurement |chapter=Stem diameter |pages=13ff |location=New York |publisher=Springer |isbn=9783540403906 }}

Image:Area-equivalent radius and diameter.svg

Image:Wetted Perimeter.svg, 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 $\backslash pi\; R^2$. Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting

:$A\; =\; \backslash pi\; R^2\_\backslash text\{eq\}$

or, alternatively:

:$R\_\backslash text\{eq\}\; =\; \backslash sqrt\{\backslash frac\{A\}\{\backslash pi\}\}$

Often the area considered is that of a cross section.

For example, a square of side length L has an area of $L^2$. Setting that area to be equal that of a circle imply that

:$R\_\backslash text\{eq\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{\backslash pi\}\}\; L\; \backslash approx\; 0.3183\; L$

Similarly, an ellipse with semi-major axis $a$ and semi-minor axis $b$ has mean radius $R\_\backslash text\{eq\}=\backslash sqrt\{a\; \backslash cdot\; b\}$.

For a circle, where $a=b$, this simplifies to $R\_\backslash text\{eq\}=a$.

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

:$D\_\backslash text\{H\}\; =\; \backslash frac\{4\; \backslash pi\; R^2\}\{2\; \backslash pi\; R\}\; =\; 2R$

: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 $D\_\backslash text\{H\}\; =\; 4\; R\_\backslash text\{H\}$, and the hydraulic radius is half of the 2D mean radius.{{cite journal | last1=Wei | first1=Maoxing | last2=Cheng | first2=Nian-Sheng | last3=Lu | first3=Yesheng | date=October 2023 | title=Revisiting the concept of hydraulic radius | journal=Journal of Hydrology | volume=625 |issue=Part B |page=130134 | doi=10.1016/j.jhydrol.2023.130134 | bibcode=2023JHyd..62530134W }}

• In aggregate classification, the equivalent diameter is the "diameter of a circle with an equal aggregate sectional area", which is calculated by $D\; =\; 2\; \backslash sqrt\{\backslash frac\{A\}\{\backslash pi\}\}$. It is used in many digital image processing programs.{{cite book | doi=10.1016/B978-0-12-849908-5.00013-4 | chapter=Asphalt mix homogeneity | title=Structural Behavior of Asphalt Pavements | date=2016 | last1=Sun | first1=Lijun | pages=821–921 | isbn=978-0-12-849908-5 }}

{{further|Mean radius (astronomy)|Equivalent spherical diameter}}

File:Ellipsoide.svg

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

:$V\; =\; \backslash frac\{4\}\{3\}\backslash pi\; R^3\_\backslash text\{eq\}$

or, alternatively:

:$R\_\backslash text\{eq\}\; =\; \backslash sqrt[3]\{\backslash frac\{3V\}\{4\backslash pi\}\}$

For example, a cube of side length L has a volume of $L^3$. Setting that volume to be equal that of a sphere imply that

:$R\_\backslash text\{eq\}\; =\; \backslash sqrt[3]\{\backslash frac\{3\}\{4\backslash pi\}\}\; L\; \backslash approx\; 0.6204\; L$

Similarly, a tri-axial ellipsoid with axes $a$, $b$ and $c$ has mean radius $R\_\backslash text\{eq\}=\backslash sqrt[3]\{a\; \backslash cdot\; b\; \backslash cdot\; c\}$.{{cite journal|url=https://www.aanda.org/articles/aa/pdf/2011/04/aa15811-10.pdf|title=Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination|first1=J.|last1=Leconte|first2=D.|last2=Lai|first3=G.|last3=Chabrier|journal=Astronomy & Astrophysics|volume=528|issue=A41|year=2011|pages=9|doi=10.1051/0004-6361/201015811|arxiv=1101.2813 |bibcode=2011A&A...528A..41L }} The formula for a rotational ellipsoid is the special case where $a=b$. Likewise, an oblate spheroid or rotational ellipsoid with axes $a$ and $c$ has a mean radius of $R\_\backslash text\{eq\}=\backslash sqrt[3]\{a^\{2\}\; \backslash cdot\; c\; \}$. For a sphere, where $a=b=c$, this simplifies to $R\_\backslash text\{eq\}=a$.

Applications:

• For planet Earth, which can be approximated as an oblate spheroid with radii {{val|6378.1|u=km}} and {{val|6356.8|u=km}}, the 3D mean radius is $R=\backslash sqrt[3]\{6378.1^\{2\}\backslash cdot6356.8\}=6371.0\backslash text\{\; km\}$.{{cite journal|url=http://frederic.chambat.free.fr/geophy/inertie_pepi01/chambat_valette_publie01_with_errata.pdf.pdf|title=Mean radius, mass, and inertia for reference Earth models|first1=F.|last1=Chambat|first2=B.|last2=Valette|journal=Physics of the Earth and Planetary Interiors|volume=124|issue=3–4|year=2001|page=4|doi=10.1016/S0031-9201(01)00200-X|bibcode=2001PEPI..124..237C }}

{{See also|authalic radius}}

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

:$4\backslash pi\; R^2\_\backslash text\{eq\}\; =\; A$

or equivalently

:$R\_\backslash text\{eq\}\; =\; \backslash sqrt\{\backslash frac\{A\}\{4\backslash pi\}\}$

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

:$R\_\backslash text\{eq\}\; =\; \backslash sqrt\{\backslash frac\{6L^2\}\{4\backslash pi\}\}=\; 0.6910\; L$

File:Osculating circle.svg

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

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

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Category:Radii

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