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.
Measurement of tree circumference, the tape calibrated to show diameter, at breast height. The tape assumes a circular shape.
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]
The area-equivalent radius of a 2D object is the radius of a circle with the same area as the objectCross 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.
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
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 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
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