# 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, , 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:

where is volume of the particle and 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.

## Ellipsoidal objects

The sphericity, , of an oblate spheroid (similar to the shape of the planet Earth) is:

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, in terms of the volume of the particle,

therefore

hence we define as:

## Sphericity of common objects

Name Picture Volume Surface Area Sphericity
Platonic Solids
tetrahedron
cube (hexahedron)

octahedron

dodecahedron

icosahedron
Round Shapes
ideal cone

hemisphere
(half sphere)

ideal cylinder

ideal torus

sphere