A capsule (from

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

'' capsula'', "small box or chest"), or

stadium A stadium ( : stadiums or stadia) is a place or venue for (mostly) outdoor sports, concerts, or other events and consists of a field or stage either partly or completely surrounded by a tiered structure designed to allow spectators to stand o ...

of revolution, is a basic three-dimensional

geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

consisting of a

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

with

hemispherical A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

ends. Another name for this shape is spherocylinder. It can also be referred to as an oval although the sides (either vertical or horizontal) are straight

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

Usages

The shape is used for some objects like containers for pressurised gases,

window A window is an opening in a wall, door, roof, or vehicle that allows the exchange of light and may also allow the passage of sound and sometimes air. Modern windows are usually glazed or covered in some other transparent or translucent materia ...

s of places like a

jet Jet, Jets, or The Jet(s) may refer to: Aerospace * Jet aircraft, an aircraft propelled by jet engines ** Jet airliner ** Jet engine ** Jet fuel * Jet Airways, an Indian airline * Wind Jet (ICAO: JET), an Italian airline * Journey to Enceladus a ...

, software buttons, building domes (like the

U.S. Capitol The United States Capitol, often called The Capitol or the Capitol Building, is the Seat of government, seat of the Legislature, legislative branch of the Federal government of the United States, United States federal government, which is form ...

, 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 Omnidirectional refers to the notion of existing in every direction. Omnidirectional devices include: * Omnidirectional antenna, an antenna that radiates equally in all directions * VHF omnidirectional range, a type of radio navigation system for ...

pattern),

mirror A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...

s, and pharmaceutical capsules. In

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

and

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, this shape is used as a basic model for non-spherical particles. It appears, in particular as a model for the molecules in

liquid crystal Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. T ...

s or for the particles in

granular matter A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). The constituents that compose gra ...

Formulas

The volume

V

of a capsule is calculated by adding the volume of a ball of radius

r

(that accounts for the two hemispheres) to the volume of the cylindrical part. Hence, if the cylinder has height

h

, :

V = \frac\pi r^3 + (\pi r^2h)= \pi r^2 \left (\fracr + h \right )

. The surface area of a capsule of radius

r

whose cylinder part has height

h

2 \pi r (2r + h)

Generalization

A capsule can be equivalently described as the Minkowski sum of a ball of radius

r

with a

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

of length

a

. By this description, capsules can be straightforwardly generalized as Minkowski sums of a ball with a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

. The resulting shape is called a spheropolyhedron.

Related shapes

A capsule is the three-dimensional shape obtained by revolving the two-dimensional

around the line of symmetry that bisects the semicircles.

References

Elementary shapes {{elementary-geometry-stub