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Equivalent radius

Radius of a circle or sphere equivalent to a non-circular or non-spherical object

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Source updated March 9, 2026

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

01Perimeter equivalent

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

$P=2\pi R_{\text{eq}}$

or, alternatively:

$R_{\text{eq}}={\frac {P}{2\pi }}$

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

$R_{\text{eq}}={\frac {4L}{2\pi }}={\frac {2L}{\pi }}\approx 0.6366L$

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.

02Area equivalent

The area of a circle of radius R is $\pi R^{2}$ . Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting

$A=\pi R_{\text{eq}}^{2}$

or, alternatively:

$R_{\text{eq}}={\sqrt {\frac {A}{\pi }}}$

Often the area considered is that of a cross section.

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

$R_{\text{eq}}={\sqrt {\frac {L^{2}}{\pi }}}={\sqrt {\frac {1}{\pi }}}L\approx 0.5642L$

Similarly, an ellipse with semi-major axis $a$ and semi-minor axis $b$ has area of $\pi ab$ , and therefore

$R_{\text{eq}}={\sqrt {\frac {\pi ab}{\pi }}}={\sqrt {ab}}$ .

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

$D_{\text{H}}={\frac {4\pi R^{2}}{2\pi R}}=2R$

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 $D_{\text{H}}=4R_{\text{H}}$ , 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 $D=2{\sqrt {\frac {A}{\pi }}}$ . It is used in many digital image processing programs.

03Volume equivalent

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

$V={\frac {4}{3}}\pi R_{\text{eq}}^{3}$

or, alternatively:

$R_{\text{eq}}={\sqrt[{3}]{\frac {3V}{4\pi }}}$

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

$R_{\text{eq}}={\sqrt[{3}]{\frac {3L^{3}}{4\pi }}}={\sqrt[{3}]{\frac {3}{4\pi }}}L\approx 0.6204L$

Similarly, a tri-axial ellipsoid with axes $a$ , $b$ and $c$ has a volume of ${\frac {4}{3}}\pi abc$ , and therefore

$R_{\text{eq}}={\sqrt[{3}]{\frac {3{\frac {4}{3}}\pi abc}{4\pi }}}={\sqrt[{3}]{abc}}$ .

The formula for a rotational ellipsoid is the special case where $a=b$

$R_{\text{eq}}={\sqrt[{3}]{a^{2}\cdot c}}$ .

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 $R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}$ .

A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right)

04Other equivalences

Surface-area equivalent radius

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

$4\pi R_{\text{eq}}^{2}=A$

or equivalently

$R_{\text{eq}}={\sqrt {\frac {A}{4\pi }}}$

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

$R_{\text{eq}}={\sqrt {\frac {6L^{2}}{4\pi }}}\approx 0.6910L$

Curvature-equivalent radius

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

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Sources and credits

This article is adapted from the Wikipedia article “Equivalent radius”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Using a DTApe.JPG by VTmaddogVT, Public domain
Area-equivalent radius and diameter.svg by Goran_tek-en, CC BY-SA 4.0
Wetted Perimeter.svg by TaylorSteiner (talk), CC BY-SA 3.0
Ellipsoide.svg by Ag2gaeh, CC BY-SA 4.0
Osculating circle.svg by Cepheus, Public domain

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Equivalent radius

01Perimeter equivalent

02Area equivalent

03Volume equivalent

04Other equivalences

Surface-area equivalent radius

Curvature-equivalent radius

Sources and credits

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