Equivalent radius


In applied sciences, the equivalent 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 is twice the equivalent radius.

Perimeter equivalent

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.
  • 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. 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 area of, and therefore
Applications:
The volume of a sphere of radius R is. 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, and has a volume of, and therefore
The formula for a rotational ellipsoid is the special case where
Applications:

Surface-area equivalent radius

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

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.