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 settingor, 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 settingor, 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 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
- 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.
Volume equivalent
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 settingor, 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:
- For planet Earth, which can be approximated as an oblate spheroid with radii and, the 3D mean radius is.
Other equivalences
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 settingor equivalently
For example, a cube of length L has a surface area of. A cube therefore has an surface area-equivalent radius of