Sphericity

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Measure of how closely a shape resembles a sphere

title: "Sphericity" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["geometric-measurement", "spheres", "metrology"] description: "Measure of how closely a shape resembles a sphere" topic_path: "general/geometric-measurement" source: "https://en.wikipedia.org/wiki/Sphericity" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Measure of how closely a shape resembles a sphere ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/c1/Rounding_&_sphericity_EN.svg" caption="rounding]] (horizontal)."] ::

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.

Definition

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

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

where V_p is volume of the object and 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.

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 object divided by the actual surface area of the object.

First we need to write surface area of the sphere, A_s in terms of the volume of the object being measured, 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

::data[format=table]

NamePictureVolumeSurface areaSphericity
Sphere[[File:Sphere wireframe 10deg 6r.svg50px]]\frac{4\pi}{3},r^34\pi,r^2
Disdyakis triacontahedron[[File:Disdyakistriacontahedron.jpg50px]]\frac{900+720\sqrt{5}}{11},s^3\frac{180\sqrt{179-24\sqrt{5}}}{11},s^2
Tricylinder[[File:Tricylinder.png50px]]16-8\sqrt{2},r^348-24\sqrt{2},r^2
Rhombic triacontahedron[[File:Rhombictriacontahedron.svg50px]]4\sqrt{5+2\sqrt{5}},s^312\sqrt{5},s^2
Icosahedron[[File:Icosahedron.svg50px]]\frac{15+5\sqrt{5}}{12},s^35\sqrt{3},s^2
Bicylinder[[File:Steinmetz-solid.svg50px]]\frac{16}{3},r^316,r^2
Ideal bicone
(h=r\sqrt{2})[[File:Bicone.svg50px]]\frac{2\pi}{3},r^{2}h=\frac{2\pi\sqrt{2}}{3},r^32\pi,r\sqrt{r^{2}+h^{2}}=2\pi\sqrt{3},r^2
Dodecahedron[[File:POV-Ray-Dodecahedron.svg50px]]\frac{15+\sqrt{5}}{4},s^33\sqrt{25+10\sqrt{5}}, s^2
Rhombic dodecahedron[[File:Rhombicdodecahedron.jpg50px]]\frac{16\sqrt{3}}{9},s^38\sqrt{2},s^2
Ideal torus(R=r)[[File:Torus2.png50px]]2\pi^2Rr^2=2\pi^2,r^34\pi^2Rr=4\pi^2,r^2
Ideal cylinder(h=2r)[[File:Circular cylinder rh.svg50px]]\pi,r^2h=2\pi,r^32\pi,r(r+h)=6\pi,r^2
Octahedron[[File:Octahedron.svg50px]]\frac{\sqrt{2}}{3},s^32\sqrt{3},s^2
Hemisphere[[File:Sphere symmetry group cs.svg50px]]\frac{2\pi}{3},r^33\pi,r^2
Cube[[File:Hexahedron.svg50px]],s^36,s^2
Ideal cone(h=2r\sqrt{2})[[File:Blue-cone.png50px]]\frac{\pi}{3},r^2h=\frac{2\pi\sqrt{2}}{3},r^3\pi,r(r+\sqrt{r^2+h^2})=4\pi,r^2
Tetrahedron[[File:Tetrahedron.svg50px]]\frac{\sqrt{2}}{12},s^3\sqrt{3},s^2
::

References

References

  1. Wadell, Hakon. (1935). "Volume, Shape, and Roundness of Quartz Particles". The Journal of Geology.

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