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.
