Sphericity

Sphericity is a measure of how closely the shape of a physical 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.

Defined by Wadell in 1935,¹ the sphericity, ${\displaystyle \Psi }$, of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:

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

where ${\displaystyle V_{p}}$ is volume of the object and ${\displaystyle A_{p}}$ 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.

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.

Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the object divided by the actual surface area of the object.

First we need to write surface area of the sphere, ${\displaystyle A_{s}}$ in terms of the volume of the object being measured, ${\displaystyle V_{p}}$

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

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:

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

Name	Picture	Volume	Surface area	Sphericity
Sphere		${\displaystyle {\frac {4\pi }{3}}\,r^{3}}$	${\displaystyle 4\pi \,r^{2}}$	${\displaystyle 1}$
Disdyakis triacontahedron		${\displaystyle {\frac {900+720{\sqrt {5}}}{11}}\,s^{3}}$	${\displaystyle {\frac {180{\sqrt {179-24{\sqrt {5}}}}}{11}}\,s^{2}}$	${\displaystyle {\frac {\left(\left(5+4{\sqrt {5}}\right)^{2}{\frac {11\pi }{5}}\right)^{\frac {1}{3}}}{\sqrt {179-24{\sqrt {5}}}}}\approx 0.9857}$
Tricylinder		${\displaystyle 16-8{\sqrt {2}}\,r^{3}}$	${\displaystyle 48-24{\sqrt {2}}\,r^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{36\pi +18\pi {\sqrt {2}}}}{6}}\approx 0.9633}$
Rhombic triacontahedron		${\displaystyle 4{\sqrt {5+2{\sqrt {5}}}}\,s^{3}}$	${\displaystyle 12{\sqrt {5}}\,s^{2}}$	${\displaystyle {\frac {\sqrt[{6}]{455625\pi ^{2}+202500\pi ^{2}{\sqrt {5}}}}{15}}\approx 0.9609}$
Icosahedron		${\displaystyle {\frac {15+5{\sqrt {5}}}{12}}\,s^{3}}$	${\displaystyle 5{\sqrt {3}}\,s^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{2100\pi {\sqrt {3}}+900\pi {\sqrt {15}}}}{30}}\approx 0.9393}$
Bicylinder		${\displaystyle {\frac {16}{3}}\,r^{3}}$	${\displaystyle 16\,r^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{2\pi }}{2}}\approx 0.9226}$
Ideal bicone ${\displaystyle (h=r{\sqrt {2}})}$		${\displaystyle {\frac {2\pi }{3}}\,r^{2}h={\frac {2\pi {\sqrt {2}}}{3}}\,r^{3}}$	${\displaystyle 2\pi \,r{\sqrt {r^{2}+h^{2}}}=2\pi {\sqrt {3}}\,r^{2}}$	${\displaystyle {\frac {\sqrt[{6}]{432}}{3}}\approx 0.9165}$
Dodecahedron		${\displaystyle {\frac {15+7{\sqrt {5}}}{4}}\,s^{3}}$	${\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}}$	${\displaystyle {\frac {{\sqrt[{6}]{2080+928{\sqrt {5}}}}{\sqrt[{3}]{9\pi }}{\sqrt {5}}}{30}}\approx 0.9105}$
Rhombic dodecahedron		${\displaystyle {\frac {16{\sqrt {3}}}{9}}\,s^{3}}$	${\displaystyle 8{\sqrt {2}}\,s^{2}}$	${\displaystyle {\frac {\sqrt[{6}]{2592\pi ^{2}}}{6}}\approx 0.9047}$
Ideal torus ${\displaystyle (R=r)}$		${\displaystyle 2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}}$	${\displaystyle 4\pi ^{2}Rr=4\pi ^{2}\,r^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{18\pi ^{2}}}{2\pi }}\approx 0.8947}$
Ideal cylinder ${\displaystyle (h=2r)}$		${\displaystyle \pi \,r^{2}h=2\pi \,r^{3}}$	${\displaystyle 2\pi \,r(r+h)=6\pi \,r^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{18}}{3}}\approx 0.8736}$
Octahedron		${\displaystyle {\frac {\sqrt {2}}{3}}\,s^{3}}$	${\displaystyle 2{\sqrt {3}}\,s^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{3\pi {\sqrt {3}}}}{3}}\approx 0.8456}$
Hemisphere		${\displaystyle {\frac {2\pi }{3}}\,r^{3}}$	${\displaystyle 3\pi \,r^{2}}$	${\displaystyle {\frac {2{\sqrt[{3}]{2}}}{3}}\approx 0.8399}$
Cube		${\displaystyle \,s^{3}}$	${\displaystyle 6\,s^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{36\pi }}{6}}\approx 0.8060}$
Ideal cone ${\displaystyle (h=2r{\sqrt {2}})}$		${\displaystyle {\frac {\pi }{3}}\,r^{2}h={\frac {2\pi {\sqrt {2}}}{3}}\,r^{3}}$	${\displaystyle \pi \,r(r+{\sqrt {r^{2}+h^{2}}})=4\pi \,r^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{4}}{2}}\approx 0.7937}$
Tetrahedron		${\displaystyle {\frac {\sqrt {2}}{12}}\,s^{3}}$	${\displaystyle {\sqrt {3}}\,s^{2}}$	${\displaystyle {\frac {\sqrt[{3}]{12\pi {\sqrt {3}}}}{6}}\approx 0.6711}$