three-dimensional geometry Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inter ...

, the girth of a geometric object, in a certain direction, is the

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

of its

parallel projection In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or ''image plane'', where the '' rays'', known as '' ...

in that direction... For instance, the girth of a

unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term '' ...

in a direction parallel to one of the three coordinate axes is four: it projects to a

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...

, which has four as its perimeter.

Surfaces of constant girth

The girth of a

sphere 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 ...

in any direction equals the

circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...

of its

equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...

, or of any of its

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...

s. More generally, if is a

surface of constant width A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

, then every projection of is a

curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane (geometry), plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of ...

, with the same width . All curves of constant width have the same perimeter, the same value as the circumference of a circle with that width (this is Barbier's theorem). Therefore, every surface of constant width is also a surface of constant girth: its girth in all directions is the same number .

Hermann Minkowski Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...

proved, conversely, that every convex surface of constant girth is also a surface of constant width.

Projection versus cross-section

For a

prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...

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 ...

, its projection in the direction parallel to its axis is the same as its

cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture and engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **A ...

, so in these cases the girth also equals the perimeter of the cross section. In some application areas such as

shipbuilding Shipbuilding is the construction of ships and other Watercraft, floating vessels. In modern times, it normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation th ...

this alternative meaning, the perimeter of a cross section, is taken as the definition of girth.

Application

Girth is sometimes used by postal services and delivery companies as a basis for pricing. For example,

Canada Post Canada Post Corporation (, trading as Canada Post (), is a Canadian Crown corporation that functions as the primary postal operator in Canada. Originally known as Royal Mail Canada (the operating name of the Post Office Department of the Can ...

requires that an item's length plus girth not exceed a maximum allowed value. For a rectangular box, the girth is 2 * (height + width), i.e. the perimeter of a projection or cross section perpendicular to its length.

References

{{reflist Euclidean solid geometry