# 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, ${\displaystyle \Psi }$, 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:

${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}}$

where ${\displaystyle V_{p}}$ is volume of the particle and ${\displaystyle A_{p}}$ 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, ${\displaystyle \Psi }$, of an oblate spheroid (similar to the shape of the planet Earth) is:

${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}},}$

where a and b are the 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, ${\displaystyle A_{s}}$ in terms of the volume of the particle, ${\displaystyle V_{p}}$

${\displaystyle A_{s}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi ^{3}r^{6}=4\pi \left(4^{2}\pi ^{2}r^{6}\right)=4\pi \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{p}^{2}}$

therefore

${\displaystyle A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}$

hence we define ${\displaystyle \Psi }$ as:

${\displaystyle \Psi ={\frac {A_{s}}{A_{p}}}={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}}$

## Sphericity of common objects

Name Picture Volume Surface Area Sphericity
Platonic Solids
tetrahedron ${\displaystyle {\frac {\sqrt {2}}{12}}\,s^{3}}$${\displaystyle {\sqrt {3}}\,s^{2}}$${\displaystyle \left({\frac {\pi }{6{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.671}$
cube (hexahedron) ${\displaystyle \,s^{3}}$${\displaystyle 6\,s^{2}}$

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

octahedron ${\displaystyle {\frac {1}{3}}{\sqrt {2}}\,s^{3}}$${\displaystyle 2{\sqrt {3}}\,s^{2}}$

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

dodecahedron ${\displaystyle {\frac {1}{4}}\left(15+7{\sqrt {5}}\right)\,s^{3}}$${\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}}$

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

icosahedron ${\displaystyle {\frac {5}{12}}\left(3+{\sqrt {5}}\right)\,s^{3}}$${\displaystyle 5{\sqrt {3}}\,s^{2}}$${\displaystyle \left({\frac {\left(3+{\sqrt {5}}\right)^{2}\pi }{60{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.939}$
Round Shapes
ideal cone
${\displaystyle (h=2{\sqrt {2}}r)}$
${\displaystyle {\frac {1}{3}}\pi \,r^{2}h}$

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

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

${\displaystyle =4\pi \,r^{2}}$

${\displaystyle \left({\frac {1}{2}}\right)^{\frac {1}{3}}\approx 0.794}$
hemisphere
(half sphere)
${\displaystyle {\frac {2}{3}}\pi \,r^{3}}$${\displaystyle 3\pi \,r^{2}}$

${\displaystyle \left({\frac {16}{27}}\right)^{\frac {1}{3}}\approx 0.840}$

ideal cylinder
${\displaystyle (h=2\,r)}$
${\displaystyle \pi r^{2}h=2\pi \,r^{3}}$${\displaystyle 2\pi r(r+h)=6\pi \,r^{2}}$

${\displaystyle \left({\frac {2}{3}}\right)^{\frac {1}{3}}\approx 0.874}$

ideal torus
${\displaystyle (R=r)}$
${\displaystyle 2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}}$${\displaystyle 4\pi ^{2}Rr=4\pi ^{2}\,r^{2}}$

${\displaystyle \left({\frac {9}{4\pi }}\right)^{\frac {1}{3}}\approx 0.894}$

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

${\displaystyle 1\,}$

Other Shapes
rhombic triacontahedron ${\displaystyle 4{\sqrt {5+2{\sqrt {5}}}}\,s^{3}}$${\displaystyle 12{\sqrt {5}}\,s^{2}}$${\displaystyle {\frac {\pi ^{\frac {1}{3}}\left(24{\sqrt {5+2{\sqrt {5}}}}\,s^{3}\right)^{\frac {2}{3}}}{12{\sqrt {5}}\,s^{2}}}\approx 0.9609}$
disdyakis triacontahedron ${\displaystyle {\frac {180}{11}}{\sqrt {179-24{\sqrt {5}}}}}$${\displaystyle {\frac {180}{11}}\left(5+4{\sqrt {5}}\right)}$${\displaystyle {\frac {\left(\left(5+4{\sqrt {5}}\right)^{2}{\frac {11\pi }{5}}\right)^{\frac {1}{3}}}{\sqrt {179-24{\sqrt {5}}}}}\approx 0.9857}$

## References

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.