- Source: Equivalent radius
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) (
D
{\displaystyle D}
) is twice the equivalent radius.
Perimeter equivalent
The perimeter of a circle 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
mean
{\displaystyle P=2\pi R_{\text{mean}}}
or, alternatively:
R
mean
=
P
2
π
{\displaystyle R_{\text{mean}}={\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
mean
=
2
L
π
≈
0.6366
L
{\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.
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.
Area equivalent
The area of a circle 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
mean
2
{\displaystyle A=\pi R_{\text{mean}}^{2}}
or, alternatively:
R
mean
=
A
π
{\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
L
2
{\displaystyle L^{2}}
. Setting that area to be equal that of a circle imply that
R
mean
=
1
π
L
≈
0.3183
L
{\displaystyle R_{\text{mean}}={\sqrt {\frac {1}{\pi }}}L\approx 0.3183L}
Similarly, an ellipse with semi-major axis
a
{\displaystyle a}
and semi-minor axis
b
{\displaystyle b}
has mean radius
R
mean
=
a
⋅
b
{\displaystyle R_{\text{mean}}={\sqrt {a\cdot b}}}
.
For a circle, where
a
=
b
{\displaystyle a=b}
, this simplifies to
R
mean
=
a
{\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
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 hydraulic radius 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_{\mathbb {H} }}
, and the hydraulic radius is half of the 2D mean radius.
In aggregate 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.
Volume equivalent
The volume of a sphere 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
mean
3
{\displaystyle V={\frac {4}{3}}\pi R_{\text{mean}}^{3}}
or, alternatively:
R
mean
=
3
V
4
π
3
{\displaystyle R_{\text{mean}}={\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
mean
=
3
4
π
3
L
≈
0.6204
L
{\displaystyle R_{\text{mean}}={\sqrt[{3}]{\frac {3}{4\pi }}}L\approx 0.6204L}
Similarly, a tri-axial ellipsoid with axes
a
{\displaystyle a}
,
b
{\displaystyle b}
and
c
{\displaystyle c}
has mean radius
R
mean
=
a
⋅
b
⋅
c
3
{\displaystyle R_{\text{mean}}={\sqrt[{3}]{a\cdot b\cdot c}}}
. The formula for a rotational ellipsoid is the special case where
a
=
b
{\displaystyle a=b}
. Likewise, an oblate spheroid or rotational ellipsoid with axes
a
{\displaystyle a}
and
c
{\displaystyle c}
has a mean radius of
R
mean
=
a
2
⋅
c
3
{\displaystyle R_{\text{mean}}={\sqrt[{3}]{a^{2}\cdot c}}}
. For a sphere, where
a
=
b
=
c
{\displaystyle a=b=c}
, this simplifies to
R
mean
=
a
{\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
R
=
6378.1
2
⋅
6356.8
3
=
6371.0
km
{\displaystyle R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}}
.
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
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