Sphericity
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. Defined by Wadell in 1935,^{[1]} the sphericity, , of a particle is the ratio of the surface area of a sphere with the same volume as the given particle to the surface area of the particle:
where is volume of the particle and is the surface area of the particle. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any particle which is not a sphere will have sphericity less than 1.
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.
Ellipsoidal objectsEdit
The sphericity, , of an oblate spheroid (similar to the shape of the planet Earth) is:
where a and b are the semimajor and semiminor axes respectively.
DerivationEdit
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, in terms of the volume of the particle,
therefore
hence we define as:
Sphericity of common objectsEdit
Name  Picture  Volume  Surface Area  Sphericity 

Platonic Solids  
tetrahedron  
cube (hexahedron) 
 
octahedron 
 
dodecahedron 
 
icosahedron  
Round Shapes  
ideal cone 



hemisphere (half sphere) 
 
ideal cylinder 
 
ideal torus 
 
sphere 
 
Other Shapes  
disdyakis triacontahedron 
See alsoEdit
ReferencesEdit
 ^ Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". The Journal of Geology. 43 (3): 250–280. Bibcode:1935JG.....43..250W. doi:10.1086/624298.
External linksEdit
Look up sphericity in Wiktionary, the free dictionary. 