Sphericity is a measure of how closely the shape of aphysical object resembles that of a perfectsphere. For example, the sphericity of theballs inside aball bearing determines thequality 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 acompactness measure of a shape.

Sphericity applies inthree dimensions; its analogue intwo dimensions, such as thecross sectional circles along acylindrical object such as ashaft, is calledroundness.

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

${\displaystyle \mathrm{\Psi }=\frac{{\pi }^{\frac{1}{3}}(6V_p{)}^{\frac{2}{3}}}{A_p}}$

where${\displaystyle V_p}$ is volume of the object and${\displaystyle A_p}$ is the surface area. The sphericity of a sphere isunity by definition and, by theisoperimetric inequality, any shape which is not a sphere will have sphericity less than 1.

The sphericity,${\displaystyle \mathrm{\Psi }}$, of anoblate spheroid (similar to the shape of the planetEarth) is:

${\displaystyle \mathrm{\Psi }=\frac{{\pi }^{\frac{1}{3}}(6V_p{)}^{\frac{2}{3}}}{A_p}=\frac{2\sqrt[3]{a{b}^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln \left(\frac{a+\sqrt{a^2-b^2}}{b}\right)},}$

wherea andb are thesemi-major andsemi-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${\displaystyle \mathrm{\Psi }}$ as:

${\displaystyle \mathrm{\Psi }=\frac{A_s}{A_p}=\frac{{\pi }^{\frac{1}{3}}{\left(6V_p\right)}^{\frac{2}{3}}}{A_p}}$

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		${\displaystyle \frac{16}{3}{r}^3}$	${\displaystyle 16r^2}$	${\displaystyle \frac{\sqrt[3]{2\pi }}{2}\approx 0.9226}$
Idealbicone ${\displaystyle (h=r\sqrt{2})}$		${\displaystyle \frac{2\pi }{3}{r}^{2}h=\frac{2\pi \sqrt{2}}{3}{r}^3}$	${\displaystyle 2\pi r\sqrt{r^2+h^2}=2\pi \sqrt{3}{r}^2}$	${\displaystyle \frac{\sqrt[6]{432}}{3}\approx 0.9165}$
Dodecahedron		${\displaystyle \frac{15+\sqrt{5}}{4}{s}^3}$	${\displaystyle 3\sqrt{25+10\sqrt{5}}{s}^2}$	${\displaystyle \frac{\sqrt[6]{2080+928\sqrt{5}}\sqrt[3]{9\pi }\sqrt{5}}{30}\approx 0.9105}$
Rhombic dodecahedron		${\displaystyle \frac{16\sqrt{3}}{9}{s}^3}$	${\displaystyle 8\sqrt{2}{s}^2}$	${\displaystyle \frac{\sqrt[6]{2592{\pi }^2}}{6}\approx 0.9047}$
Idealtorus ${\displaystyle (R=r)}$		${\displaystyle 2{\pi }^{2}R{r}^2=2{\pi }^2{r}^3}$	${\displaystyle 4{\pi }^{2}Rr=4{\pi }^2{r}^2}$	${\displaystyle \frac{\sqrt[3]{18{\pi }^2}}{2\pi }\approx 0.8947}$
Idealcylinder ${\displaystyle (h=2r)}$		${\displaystyle \pi r^{2}h=2\pi r^3}$	${\displaystyle 2\pi r(r+h)=6\pi r^2}$	${\displaystyle \frac{\sqrt[3]{18}}{3}\approx 0.8736}$
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		${\displaystyle s^3}$	${\displaystyle 6s^2}$	${\displaystyle \frac{\sqrt[3]{36\pi }}{6}\approx 0.8060}$
Idealcone ${\displaystyle (h=2r\sqrt{2})}$		${\displaystyle \frac{\pi }{3}{r}^{2}h=\frac{2\pi \sqrt{2}}{3}{r}^3}$	${\displaystyle \pi r(r+\sqrt{r^2+h^2})=4\pi r^2}$	${\displaystyle \frac{\sqrt[3]{4}}{2}\approx 0.7937}$
Tetrahedron		${\displaystyle \frac{\sqrt{2}}{12}{s}^3}$	${\displaystyle \sqrt{3}{s}^2}$	${\displaystyle \frac{\sqrt[3]{12\pi \sqrt{3}}}{6}\approx 0.6711}$

• Equivalent spherical diameter
• Flattening
• Isoperimetric ratio
• Rounding (sediment)
• Roundness
• Willmore energy

Look upsphericity in Wiktionary, the free dictionary.

• Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

• Geometric measurement
• Spheres
• Metrology

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