Equivalent radius

<h2 id="perimeter-equivalent">Perimeter equivalent</h2>

The <a href="/facts/Perimeter_of_a_circle/r1rntU0f">perimeter of a circle</a> of radius R is 
 
 
 
 2
 π
 R
 
 
 {\displaystyle 2\pi R}
 
. Given the perimeter of a non-circular object P, one can calculate its perimeter-equivalent radius by setting

P
        =
        2
        π
        
          R
          
            eq
          
        
      
    
    {\displaystyle P=2\pi R_{\text{eq}}}

or, alternatively:

R
          
            eq
          
        
        =
        
          
            P
            
              2
              π
            
          
        
      
    
    {\displaystyle R_{\text{eq}}={\frac {P}{2\pi }}}

For example, a square of side L has a perimeter of 
 
 
 
 4
 L
 
 
 {\displaystyle 4L}
 
. Setting that perimeter to be equal to that of a circle imply that

R
          
            eq
          
        
        =
        
          
            
              2
              L
            
            π
          
        
        ≈
        0.6366
        L
      
    
    {\displaystyle R_{\text{eq}}={\frac {2L}{\pi }}\approx 0.6366L}

Applications:

<ul><li><a href="/facts/US_hat_size/EHhEd47A">US hat size</a> is the circumference of the head, measured in inches, divided by <a href="/facts/Pi_(constant)/vC0UBj1q">pi</a>, rounded to the nearest 1/8 inch. This corresponds to the 1D mean diameter.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li><a href="/facts/Diameter_at_breast_height/La9IvRNv">Diameter at breast height</a> is the circumference of <a href="/facts/Tree_trunk/13Nq8CEC">tree trunk</a>, 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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
<h2 id="area-equivalent">Area equivalent</h2>

The <a href="/facts/Area_of_a_circle/sRUeg8Cl">area of a circle</a> of radius R is 
 
 
 
 π
 
 R
 
 2
 
 
 
 
 {\displaystyle \pi R^{2}}
 
. Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting

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

or, alternatively:

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

Often the area considered is that of a <a href="/facts/Cross_section_(geometry)/g5R1c6v2">cross section</a>.
For example, a square of side length L has an area of 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L^{2}}
 
. Setting that area to be equal that of a circle imply that

R
          
            eq
          
        
        =
        
          
            
              1
              π
            
          
        
        L
        ≈
        0.3183
        L
      
    
    {\displaystyle R_{\text{eq}}={\sqrt {\frac {1}{\pi }}}L\approx 0.3183L}

Similarly, an <a href="/facts/Ellipse/1wUDnGxY">ellipse</a> with <a href="/facts/Semi-major_axis/l7JES2wo">semi-major axis</a> 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and <a href="/facts/Semi-minor_axis/l7JES2wo">semi-minor axis</a> 
 
 
 
 b
 
 
 {\displaystyle b}
 
 has mean radius 
 
 
 
 
 R
 
 eq
 
 
 =
 
 
 a
 ⋅
 b
 
 
 
 
 {\displaystyle R_{\text{eq}}={\sqrt {a\cdot b}}}
 
.
For a circle, where 
 
 
 
 a
 =
 b
 
 
 {\displaystyle a=b}
 
, this simplifies to 
 
 
 
 
 R
 
 eq
 
 
 =
 a
 
 
 {\displaystyle R_{\text{eq}}=a}
 
.
Applications:

<ul><li>The <a href="/facts/Hydraulic_diameter/kqB4DVbX">hydraulic diameter</a> is similarly defined as 4 times the <a href="/facts/Cross-sectional_area/g5R1c6v2">cross-sectional area</a> of a pipe A, divided by its <a href="/facts/Wetted_perimeter/fBF50tK8">"wetted" perimeter</a> P. For a circular pipe of radius R, at full flow, this is</li></ul>

D
          
            H
          
        
        =
        
          
            
              4
              π
              
                R
                
                  2
                
              
            
            
              2
              π
              R
            
          
        
        =
        2
        R
      
    
    {\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 <a href="/facts/Hydraulic_radius/7qc8CChC">hydraulic radius</a> is defined as the cross-sectional area of a pipe A, divided by its wetted perimeter P, which leads to 
 
 
 
 
 D
 
 H
 
 
 =
 4
 
 R
 
 H
 
 
 
 
 {\displaystyle D_{\text{H}}=4R_{\text{H}}}
 
, and the hydraulic radius is half of the 2D mean radius.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
<ul><li>In <a href="/facts/Construction_aggregate/K7XM0BAg">aggregate</a> classification, the equivalent diameter is the "diameter of a circle with an equal aggregate sectional area", which is calculated by 
 
 
 
 D
 =
 2
 
 
 
 A
 π
 
 
 
 
 
 {\displaystyle D=2{\sqrt {\frac {A}{\pi }}}}
 
. It is used in many digital image processing programs.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li></ul>
<h2 id="volume-equivalent">Volume equivalent</h2>
Further information: <a href="/facts/Mean_radius_(astronomy)/KDBqwPKg">Mean radius (astronomy)</a> and <a href="/facts/Equivalent_spherical_diameter/j4SQoe6j">Equivalent spherical diameter</a>

The <a href="/facts/Volume_of_a_sphere/jywYLNM2">volume of a sphere</a> of radius R is 
 
 
 
 
 
 4
 3
 
 
 π
 
 R
 
 3
 
 
 
 
 {\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

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

or, alternatively:

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

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

R
          
            eq
          
        
        =
        
          
            
              3
              
                4
                π
              
            
            
              3
            
          
        
        L
        ≈
        0.6204
        L
      
    
    {\displaystyle R_{\text{eq}}={\sqrt[{3}]{\frac {3}{4\pi }}}L\approx 0.6204L}

Similarly, a <a href="/facts/Tri-axial_ellipsoid/pe9vm0Sh">tri-axial ellipsoid</a> with axes 
 
 
 
 a
 
 
 {\displaystyle a}
 
, 
 
 
 
 b
 
 
 {\displaystyle b}
 
 and 
 
 
 
 c
 
 
 {\displaystyle c}
 
 has mean radius 
 
 
 
 
 R
 
 eq
 
 
 =
 
 
 
 a
 ⋅
 b
 ⋅
 c
 
 
 3
 
 
 
 
 
 {\displaystyle R_{\text{eq}}={\sqrt[{3}]{a\cdot b\cdot c}}}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> The formula for a rotational ellipsoid is the special case where 
 
 
 
 a
 =
 b
 
 
 {\displaystyle a=b}
 
. Likewise, an <a href="/facts/Oblate_spheroid/GJyyLxQ7">oblate spheroid</a> or <a href="/facts/Rotational_ellipsoid/GJyyLxQ7">rotational ellipsoid</a> with axes 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 c
 
 
 {\displaystyle c}
 
 has a mean radius of 
 
 
 
 
 R
 
 eq
 
 
 =
 
 
 
 
 a
 
 2
 
 
 ⋅
 c
 
 
 3
 
 
 
 
 
 {\displaystyle R_{\text{eq}}={\sqrt[{3}]{a^{2}\cdot c}}}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> For a sphere, where 
 
 
 
 a
 =
 b
 =
 c
 
 
 {\displaystyle a=b=c}
 
, this simplifies to 
 
 
 
 
 R
 
 eq
 
 
 =
 a
 
 
 {\displaystyle R_{\text{eq}}=a}
 
.
Applications:

<ul><li>For planet <a href="/facts/Earth/HLLmDfj0">Earth</a>, which can be approximated as an oblate spheroid with radii 6378.1 km and 6356.8 km, the 3D mean radius is 
 
 
 
 R
 =
 
 
 
 
 6378.1
 
 2
 
 
 ⋅
 6356.8
 
 
 3
 
 
 
 =
 6371.0
 
  km
 
 
 
 {\displaystyle R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>
<h2 id="other-equivalences">Other equivalences</h2>
<h3>Surface-area equivalent radius</h3>
See also: <a href="/facts/Authalic_radius/PVsf0g2I">authalic radius</a>
The <a href="/facts/Surface_area/WNowzFCw">surface area</a> of a sphere of radius R is 
 
 
 
 4
 π
 
 R
 
 2
 
 
 
 
 {\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

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

or equivalently

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

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

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

<h3>Curvature-equivalent radius</h3>

The <a href="/facts/Osculating_circle/mDs0CS3x">osculating circle</a> and <a href="/facts/Osculating_sphere/z1LvpVgd">osculating sphere</a> define <a href="/facts/Curvature/RFCAEsj8">curvature</a>-equivalent radii at a particular <a href="/facts/Point_of_tangency/sF0jH3EI">point of tangency</a> for <a href="/facts/Plane_figure/nww0x85d">plane figures</a> and solid figures, respectively.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Antenna_equivalent_radius/9n0TfrWY">Antenna equivalent radius</a></li>
<li><a href="/facts/Cloud_drop_effective_radius/cyFpx6XU">Cloud drop effective radius</a></li>
<li><a href="/facts/Cubic_mean/B9IH6dFM">Cubic mean</a></li>
<li><a href="/facts/Earth_ellipsoid/l88dGWLD">Earth ellipsoid</a></li>
<li><a href="/facts/Earth_radius/PVsf0g2I">Earth radius</a></li>
<li><a href="/facts/Galaxy_effective_radius/KngX0Vnm">Galaxy effective radius</a></li>
<li><a href="/facts/Geoid/6aR04HQl">Geoid</a></li>
<li><a href="/facts/Geometric_mean/6Nf6uk1A">Geometric mean</a></li>
<li><a href="/facts/Semidiameter/yzsFRcgJ">Semidiameter</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bello, Ignacio; Britton, Jack Rolf (1993). Topics in Contemporary Mathematics (5th ed.). Lexington, Mass: D.C. Heath. p. 512. ISBN 9780669289572. <a href="9780669289572" target="_blank">9780669289572</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">West, P. W. (2004). "Stem diameter". Tree and Forest Measurement. New York: Springer. pp. 13ff. ISBN 9783540403906. <a href="9783540403906" target="_blank">9783540403906</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-0-12-849908-5" target="_blank">978-0-12-849908-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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 href="https://www.aanda.org/articles/aa/pdf/2011/04/aa15811-10.pdf" target="_blank">https://www.aanda.org/articles/aa/pdf/2011/04/aa15811-10.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://frederic.chambat.free.fr/geophy/inertie_pepi01/chambat_valette_publie01_with_errata.pdf.pdf" target="_blank">http://frederic.chambat.free.fr/geophy/inertie_pepi01/chambat_valette_publie01_with_errata.pdf.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">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. <a href="http://frederic.chambat.free.fr/geophy/inertie_pepi01/chambat_valette_publie01_with_errata.pdf.pdf" target="_blank">http://frederic.chambat.free.fr/geophy/inertie_pepi01/chambat_valette_publie01_with_errata.pdf.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Equivalent radius open-in-new

Equivalent radius