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:
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.[1]
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.[2]
The area of a circle of radius R is . Given the area of an 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
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 , and the hydraulic radius is half of the 2D mean radius.[3]
In aggregate classification, the equivalent diameter is the "diameter of a circle with an equal aggregate sectional area", which is calculated by . It is used in many digital image processing programs.[4]
The volume of a sphere of radius R is . Given the volume of an 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 , and has mean radius .[5] The formula for a rotational ellipsoid is the special case where . Likewise, an oblate spheroid or rotational ellipsoid with axes and has a mean radius of .[6] For a sphere, where , this simplifies to .
Applications:
For planet Earth, which can be approximated as an oblate spheroid with radii 6378.1km and 6356.8km, the 3D mean radius is .[6]