A capsule (from

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

'' 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 completely or partially surrounded by a tiered structure designed to allow spectators to stand or sit ...

of revolution, is a basic three-dimensional

geometric shape A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, ''shape'' excludes informat ...

consisting of a

cylinder A cylinder () 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 infinite ...

with

hemispherical A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, 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 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 of mathematics (projective geometry, technical drawing, etc.), it is given a more precise definition, which may inc ...

although the sides (either vertical or horizontal) are straight

parallel Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science a ...

Usages

The shape is used for some objects like containers for pressurised gases, building domes, and pharmaceutical capsules. In

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

and

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

, 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 can flow like a liquid, but its molecules may be oriented in a common direction as i ...

s or for the particles in granular matter.

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 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'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

of a ball of radius

r

with a

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

of length

a

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

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

. 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 two-di ...

that bisects the semicircles.

References

Elementary shapes {{elementary-geometry-stub