Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape.
Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft, is called roundness.
Defined by Wadell in 1935, [1] the sphericity, Ψ{\displaystyle \Psi }, of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:
where Vp{\displaystyle V_{p}} is volume of the object and Ap{\displaystyle A_{p}} is the surface area. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any shape which is not a sphere will have sphericity less than 1.
The sphericity, Ψ{\displaystyle \Psi }, of an oblate spheroid (similar to the shape of the planet Earth) is:
where a and b are the semi-major and semi-minor axes respectively.
Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the object divided by the actual surface area of the object.
First we need to write surface area of the sphere, As{\displaystyle A_{s}} in terms of the volume of the object being measured, Vp{\displaystyle V_{p}}
therefore
hence we define Ψ{\displaystyle \Psi } as:
