# Girth (geometry)

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{{Short description|Perimeter of a 3D object's parallel projection in a given direction}}
In [three-dimensional geometry](/source/three-dimensional_geometry), the '''girth''' of a geometric object, in a certain direction, is the [perimeter](/source/perimeter) of its [parallel projection](/source/parallel_projection) in that direction.<ref name="gati">{{citation
 | last1 = Hilbert | first1 = David | author1-link = David Hilbert
 | last2 = Cohn-Vossen | first2 = Stephan | author2-link = Stephan Cohn-Vossen
 | edition = 2nd
 | isbn = 0-8284-1087-9
 | pages = 216–217
 | publisher = Chelsea
 | url = https://books.google.com/books?id=7WY5AAAAQBAJ&pg=PA216
 | title = Geometry and the Imagination
 | year = 1952}}.</ref><ref name="groemer">{{citation|title=Geometric Applications of Fourier Series and Spherical Harmonics|volume=61|series=Encyclopedia of Mathematics and its Applications|first=H.|last=Groemer|publisher=Cambridge University Press|year=1996|isbn=9780521473187|page=219|url=https://books.google.com/books?id=WSVyJLgJRHYC&pg=PA219}}.</ref> For instance, the girth of a [unit cube](/source/unit_cube) in a direction parallel to one of the three coordinate axes is four: it projects to a [unit square](/source/unit_square), which has four as its perimeter.

==Surfaces of constant girth==

The girth of a [sphere](/source/sphere) in any direction equals the [circumference](/source/circumference) of its [equator](/source/equator), or of any of its [great circle](/source/great_circle)s. More generally,
if {{mvar|S}} is a [surface of constant width](/source/surface_of_constant_width) {{mvar|w}}, then every projection of {{mvar|S}} is a [curve of constant width](/source/curve_of_constant_width), with the same width {{mvar|w}}. All curves of constant width have the same perimeter, the same value {{mvar|πw}} as the circumference of a circle with that width (this is [Barbier's theorem](/source/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 {{mvar|πw}}. [Hermann Minkowski](/source/Hermann_Minkowski) proved, conversely, that every convex surface of constant girth is also a surface of constant width.<ref name="gati"/><ref name="groemer"/>

==Projection versus cross-section==
For a [prism](/source/prism_(geometry)) or [cylinder](/source/Cylinder_(geometry)), its projection in the direction parallel to its axis is the same as its [cross section](/source/cross_section_(geometry)), so in these cases the girth also equals the perimeter of the cross section. In some application areas such as [shipbuilding](/source/shipbuilding) this alternative meaning, the perimeter of a cross section, is taken as the definition of girth.<ref>{{citation|title=Introduction to Naval Architecture|first=Thomas Charles|last=Gillmer|publisher=Naval Institute Press|year=1982|isbn=9780870213182|page=[https://archive.org/details/introductiontona0000gill/page/305 305]|url=https://archive.org/details/introductiontona0000gill/page/305}}.</ref>

==Application==
Girth is sometimes used by postal services and delivery companies as a basis for pricing. For example, [Canada Post](/source/Canada_Post) requires that an item's length plus girth not exceed a maximum allowed value.<ref>{{cite web
| url = http://www.canadapost.ca/tools/pg/prices/CPcanada-e.asp
| title = Canada
| publisher = [Canada Post](/source/Canada_Post)
| date = 2008-01-14
| accessdate = 2008-03-13
}}</ref> 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}}

Category:Euclidean solid geometry

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Adapted from the Wikipedia article [Girth (geometry)](https://en.wikipedia.org/wiki/Girth_(geometry)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Girth_(geometry)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.