# Sphericity

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:

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

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

$1\,$

## Sphericity in statistics

In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. A sufficient (but not necessary) condition for sphericity is that the variances of the sets of scores are equal and the covariances of the sets of scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is one of the statistical tests used to evaluate sphericity.