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 ...

'' 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 ...

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 infi ...

with

hemispherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...

ends. Another name for this shape is spherocylinder. It can also be referred to as an

oval An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one o ...

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 I ...

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 mate ...

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 the legislative branch of the United States federal government, which is formally known as the United States Congress. It is located on Capitol Hill at ...

, having the windows of the top hat that depict

The Apotheosis of Washington ''The Apotheosis of Washington'' is the fresco painted by Greek- Italian artist Constantino Brumidi in 1865 and visible through the oculus of the dome in the rotunda of the United States Capitol Building. The fresco is suspended above the ...

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 A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated li ...

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 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 rel ...

, 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. The ...

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 In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowsk ...

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 ...

. 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 In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D t ...

that bisects the semicircles.

References

Elementary shapes {{elementary-geometry-stub