Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and
rounding (horizontal).
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, $\Psi$, 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:
 $\Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}$
where $V_{p}$ is volume of the particle and $A_{p}$ 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 objects
The sphericity, $\Psi$, of an oblate spheroid (similar to the shape of the planet Earth) is:
 $\Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}b^{2}}}}{b}}\right)}}},$
where a and b are the semimajor and semiminor 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 particle, $V_{p}$
 $A_{s}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi ^{3}r^{6}=4\pi \left(4^{2}\pi ^{2}r^{6}\right)=4\pi \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{p}^{2}$
therefore
 $A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}$
hence we define $\Psi$ as:
 $\Psi ={\frac {A_{s}}{A_{p}}}={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}$
Sphericity of common objects
Name

Picture

Volume

Surface Area

Sphericity


Platonic Solids

tetrahedron


${\frac {\sqrt {2}}{12}}\,s^{3}$ 
${\sqrt {3}}\,s^{2}$ 
$\left({\frac {\pi }{6{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.671$

cube (hexahedron)


$\,s^{3}$ 
$6\,s^{2}$ 
$\left({\frac {\pi }{6}}\right)^{\frac {1}{3}}\approx 0.806$

octahedron


${\frac {1}{3}}{\sqrt {2}}\,s^{3}$ 
$2{\sqrt {3}}\,s^{2}$ 
$\left({\frac {\pi }{3{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.846$

dodecahedron


${\frac {1}{4}}\left(15+7{\sqrt {5}}\right)\,s^{3}$ 
$3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}$ 
$\left({\frac {\left(15+7{\sqrt {5}}\right)^{2}\pi }{12\left(25+10{\sqrt {5}}\right)^{\frac {3}{2}}}}\right)^{\frac {1}{3}}\approx 0.910$

icosahedron


${\frac {5}{12}}\left(3+{\sqrt {5}}\right)\,s^{3}$ 
$5{\sqrt {3}}\,s^{2}$ 
$\left({\frac {\left(3+{\sqrt {5}}\right)^{2}\pi }{60{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.939$

Round Shapes

ideal cone $(h=2{\sqrt {2}}r)$


${\frac {1}{3}}\pi \,r^{2}h$
$={\frac {2{\sqrt {2}}}{3}}\pi \,r^{3}$

$\pi \,r(r+{\sqrt {r^{2}+h^{2}}})$
$=4\pi \,r^{2}$

$\left({\frac {1}{2}}\right)^{\frac {1}{3}}\approx 0.794$

hemisphere (half sphere)


${\frac {2}{3}}\pi \,r^{3}$ 
$3\pi \,r^{2}$ 
$\left({\frac {16}{27}}\right)^{\frac {1}{3}}\approx 0.840$

ideal cylinder $(h=2\,r)$


$\pi r^{2}h=2\pi \,r^{3}$ 
$2\pi r(r+h)=6\pi \,r^{2}$ 
$\left({\frac {2}{3}}\right)^{\frac {1}{3}}\approx 0.874$

ideal torus $(R=r)$


$2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}$ 
$4\pi ^{2}Rr=4\pi ^{2}\,r^{2}$ 
$\left({\frac {9}{4\pi }}\right)^{\frac {1}{3}}\approx 0.894$

sphere


${\frac {4}{3}}\pi r^{3}$ 
$4\pi \,r^{2}$ 
$1\,$

Other Shapes

disdyakis triacontahedron


${\frac {180}{11}}{\sqrt {17924{\sqrt {5}}}}$ 
${\frac {180}{11}}\left(5+4{\sqrt {5}}\right)$ 
${\frac {\left(\left(5+4{\sqrt {5}}\right)^{2}{\frac {11\pi }{5}}\right)^{\frac {1}{3}}}{\sqrt {17924{\sqrt {5}}}}}\approx 0.986$

See also
References
External links
Look up sphericity in Wiktionary, the free dictionary. 