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.

Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

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

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Defined by Wadell in 1935,[1] the sphericity,  , of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:

 

where   is volume of the object and   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

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The sphericity,  , 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

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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 object being measured,  

 

therefore

 

hence we define   as:

 

Sphericity of common objects

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Name Picture Volume Surface area Sphericity
Sphere       1
Disdyakis triacontahedron        
Rhombic triacontahedron        
Icosahedron        
Dodecahedron        
Ideal torus
 
       
Ideal cylinder
 
       
Octahedron        
Hemisphere
(half sphere)
       
Cube (hexahedron)        
Ideal cone
 
       
Tetrahedron        

See also

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References

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  1. ^ 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.
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