Capsule (geometry)

The shape is used for some objects like containers for pressurised gases, windows of places like a jet, software buttons, building domes (like the U.S. Capitol, having the windows of the top hat that depict The Apotheosis of Washington inside designed with the appearance of the shape & placed in an omnidirectional pattern), mirrors, and pharmaceutical capsules.

In chemistry and physics, this shape is used as a basic model for non-spherical particles. It appears, in particular as a model for the molecules in liquid crystals[6][3][4] or for the particles in granular matter.[5][7][8]

The volume ${\displaystyle V}$ of a capsule is calculated by adding the volume of a ball of radius ${\displaystyle r}$ (that accounts for the two hemispheres) to the volume of the cylindrical part. Hence, if the cylinder has height ${\displaystyle h}$,

${\displaystyle V={\frac {4}{3}}\pi r^{3}+(\pi r^{2}h)=\pi r^{2}\left({\frac {4}{3}}r+h\right)}$.

The surface area of a capsule of radius ${\displaystyle r}$ whose cylinder part has height ${\displaystyle h}$ is ${\displaystyle 2\pi r(2r+h)}$.

A capsule can be equivalently described as the Minkowski sum of a ball of radius ${\displaystyle r}$ with a line segment of length ${\displaystyle a}$.[5] By this description, capsules can be straightforwardly generalized as Minkowski sums of a ball with a polyhedron. The resulting shape is called a spheropolyhedron.[7][8]

A capsule is the three-dimensional shape obtained by revolving the two-dimensional stadium around the line of symmetry that bisects the semicircles.