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.

Definition

Defined by Wadell in 1935, the sphericity, \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 V_p is volume of the object and 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.

Ellipsoidal objects

The sphericity, \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.

Derivation

Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle. First we need to write surface area of the sphere, A_s in terms of the volume of the object being measured, V_p therefore hence we define \Psi as:

Sphericity of common objects