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) (

) is twice the equivalent radius.

The perimeter of a circle of radius R is

Given the perimeter of a non-circular object P, one can calculate its perimeter-equivalent radius by setting or, alternatively: For example, a square of side L has a perimeter of

Setting that perimeter to be equal to that of a circle imply that Applications: The area of a circle of radius R is

π

Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting or, alternatively: Often the area considered is that of a cross section.

For example, a square of side length L has an area of

Setting that area to be equal that of a circle imply that Similarly, an ellipse with semi-major axis

and semi-minor axis

has mean radius

Applications: The volume of a sphere of radius R is

π

Given the volume of a non-spherical object V, one can calculate its volume-equivalent radius by setting or, alternatively: For example, a cube of side length L has a volume of

Setting that volume to be equal that of a sphere imply that Similarly, a tri-axial ellipsoid with axes

[5] The formula for a rotational ellipsoid is the special case where

Likewise, an oblate spheroid or rotational ellipsoid with axes

Applications: The surface area of a sphere of radius R is

Given the surface area of a non-spherical object A, one can calculate its surface area-equivalent radius by setting or equivalently For example, a cube of length L has a surface area of

A cube therefore has an surface area-equivalent radius of The osculating circle and osculating sphere define curvature-equivalent radii at a particular point of tangency for plane figures and solid figures, respectively.

Measurement of tree circumference, the tape calibrated to show diameter, at breast height. The tape assumes a circular shape.
The area-equivalent radius of a 2D object is the radius of a circle with the same area as the object
Cross sectional area of a trapezoidal open channel, red highlights the wetted perimeter , where water is in contact with the channel. The hydraulic diameter is the equivalent circular configuration with the same circumference as the wetted perimeter.
A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right)
An osculating circle