Equivalent radius

 ( Radii ) 

In , 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) ($D$) is twice the equivalent radius.

---

Perimeter equivalent 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.
  (1993). 9780669289572, 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.
  (2025). 9783540403906, Springer. ISBN 9783540403906

---

Area equivalent 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.

• 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.
  (2025). 9780128499085 ISBN 9780128499085

---

Volume equivalent 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:

$R\_\backslash text\{eq\}\; =\; \backslash sqrt3\{\backslash frac\{3V\}\{4\backslash pi\}\}$

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

$R\_\backslash text\{eq\}\; =\; \backslash sqrt3\{\backslash frac\{3\}\{4\backslash pi\}\}\; L\; \backslash approx\; 0.6204\; L$

Similarly, a tri-axial ellipsoid with axes $a$, $b$ and $c$ has mean radius $R\_\backslash text\{eq\}=\backslash sqrt3\{a\; \backslash cdot\; b\; \backslash cdot\; c\}$. 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 $R\_\backslash text\{eq\}=\backslash sqrt3\{a^\{2\}\; \backslash cdot\; c\; \}$. 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 and , the 3D mean radius is $R=\backslash sqrt3\{6378.1^\{2\}\backslash cdot6356.8\}=6371.0\backslash text\{\; km\}$.

---

Other equivalences

---

Surface-area equivalent 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

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

---

Curvature-equivalent radius The osculating circle and osculating sphere define curvature-equivalent radii at a particular point of tangency for 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

Post Comment