Sphericity

For sphericity in statistics, see Mauchly's sphericity test.
Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

Sphericity is a measure of how spherical (round) an object is. As such, it 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 1 and, by the isoperimetric inequality, any particle which is not a sphere will have sphericity less than 1.

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 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, $A_s$ 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 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
$(R=r)$
$2 \pi^2 R r^2 = 2 \pi^2 \,r^3$ $4 \pi^2 R r = 4 \pi^2\,r^2$

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

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

$1\,$