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{{Short description|In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron}}
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[[Image:Cube diagonals.svg|thumb|right|The diagonals of a [[cube]] with side length 1.  AC' (shown in blue) is a [[space diagonal]] with length <math>\sqrt 3</math>, while AC (shown in red) is a [[face diagonal]] and has length <math>\sqrt 2</math>.]]
[[File:Horizontal vertical and diagonal.svg|thumb|Horizontal (left), vertical (center) and diagonal (right) double arrows.]]

In [[geometry]], a '''diagonal''' is a [[line segment]] joining two [[vertex (geometry)|vertices]] of a [[polygon]] or [[polyhedron]], when those vertices are not on the same [[edge (geometry)|edge]].  Informally, any sloping line is called diagonal.  The word ''diagonal'' derives from the [[ancient Greek]] διαγώνιος ''diagonios'',<ref>{{Cite web |last=Harper |first=Douglas R. |year=2018 |title=diagonal (adj.) |url=https://www.etymonline.com/search?q=diagonal |website=[[Online Etymology Dictionary]] }}</ref> "from corner to corner" (from  διά- ''dia-'', "through", "across" and γωνία ''gonia'', "corner", related to ''gony'' "knee"); it was used by both [[Strabo]]<ref>Strabo, Geography 2.1.36–37</ref> and [[Euclid]]<ref>Euclid, Elements book 11, proposition 28</ref> to refer to a line connecting two vertices of a [[rhombus]] or [[cuboid]],<ref>Euclid, Elements book 11, proposition 38</ref> and later adopted into Latin as ''diagonus'' ("slanting line").

==Polygons==
As applied to a [[polygon]], a diagonal is a [[line segment]] joining any two non-consecutive vertices.  Therefore, a [[quadrilateral]] has two diagonals, joining opposite pairs of vertices.  For any [[convex polygon]], all the diagonals are inside the polygon, but for [[re-entrant polygon]]s, some diagonals are outside of the polygon.

Any ''n''-sided polygon (''n'' ≥ 3), [[Convex polygon|convex]] or [[Concave polygon|concave]], has <math>\tfrac{n(n-3)}{2}</math> ''total'' diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or {{math|''n'' − 3}} diagonals, and each diagonal is shared by two vertices.

In general, a regular ''n''-sided polygon has <math>\left\lfloor\frac {n-2}{2}\right\rfloor</math> ''distinct'' diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.

{|rules="none" border="0" cellspacing="4" cellpadding="0" style="background:transparent;text-align:right"
|-valign="top"
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 3 || 0
|-
| 4 || 2
|-
| 5 || 5
|-
| 6 || 9
|-
| 7 || 14
|-
| 8 || 20
|-
| 9 || 27
|-
| 10 || 35
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 11 || 44
|-
| 12 || 54
|-
| 13 || 65
|-
| 14 || 77
|-
| 15 || 90
|-
| 16 || 104
|-
| 17 || 119
|-
| 18 || 135
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 19 || 152
|-
| 20 || 170
|-
| 21 || 189
|-
| 22 || 209
|-
| 23 || 230
|-
| 24 || 252
|-
| 25 || 275
|-
| 26 || 299
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 27 || 324
|-
| 28 || 350
|-
| 29 || 377
|-
| 30 || 405
|-
| 31 || 434
|-
| 32 || 464
|-
| 33 || 495
|-
| 34 || 527
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 35 || 560
|-
| 36 || 594
|-
| 37 || 629
|-
| 38 || 665
|-
| 39 || 702
|-
| 40 || 740
|-
| 41 || 779
|-
| 42 || 819
|}
|}

===Regions formed by diagonals===

In a [[convex polygon]], if no three diagonals are [[concurrent lines|concurrent]] at a single point in the interior, the number of regions that the diagonals divide the interior into is given by<ref>{{cite book |chapter=A Problem in Combinatorics |at={{nobr|Ch. 9}}, {{pgs|99–107}} |last=Honsberger |title=Mathematical Gems |chapter-url=https://archive.org/details/mathematicalgems0001hons/page/99/?q=%22diagonals+of+a+square+divide%22 |chapter-url-access=limited |publisher=Mathematical Association of America |year=1973 |isbn=0-88385-301-9 }} {{pb}} {{cite journal |title=The Number of Regions Determined by a Convex Polygon |last=Freeman |first=J. W. |journal=Mathematics Magazine |volume=49 |number=1 |year=1976 |pages=23–25 |doi=10.2307/2689875 |jstor=2689875 }}</ref>

:<math>\binom n4 + \binom {n-1}2 = \frac{(n-1)(n-2)(n^2-3n+12)}{24}.</math>

For ''n''-gons with ''n''=3, 4, ... the number of regions is
:1, 4, 11, 25, 50, 91, 154, 246...

This is [[OEIS]] sequence A006522.<ref>{{Cite OEIS|1=A006522}}</ref>

===Intersections of diagonals===
If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by <math>\textstyle \binom n4</math>.<ref>Poonen, Bjorn; Rubinstein, Michael. "The number of intersection points made by the diagonals of a regular polygon". ''SIAM J. Discrete Math''. 11 (1998), no. 1, 135–156; [https://math.mit.edu/~poonen/papers/ngon.pdf link to a version on Poonen's website] </ref><ref name="youtube">{{Cite AV media |url=https://www.youtube.com/watch?v=K8P8uFahAgc |title=Circle Division Solution (old version) |date=2015-05-23 |last=3Blue1Brown |access-date=2024-09-01 |via=YouTube}}</ref> This holds, for example, for any [[regular polygon]] with an odd number of sides. The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the ''n'' vertices four at a time.

===Regular polygons===
{{See also|Quadrilateral#Diagonals|Hexagon#Convex equilateral hexagon|Heptagon#Diagonals and heptagonal triangle}}

Although the number of distinct diagonals in a polygon increases as its number of sides increases, the length of any diagonal can be calculated. 

In a regular n-gon with side length ''a'', the length of the ''xth'' shortest distinct diagonal is:

:<math>\sin \left(\frac{\pi(x+1)}{n}\right) \csc \left(\frac{\pi}{n}\right) \cdot a </math>

This formula shows that as the number of sides approaches infinity, the ''xth'' shortest diagonal approaches the length {{tmath|(x+1)a}}. Additionally, the formula for the shortest diagonal simplifies in the case of x = 1:

:<math>\sin \left(\frac{2\pi}{n}\right) \csc \left(\frac{\pi}{n}\right) \cdot a = 2\cos \left(\frac{\pi}{n}\right) \cdot a</math>

If the number of sides is even, the longest diagonal will be equivalent to the diameter of the polygon's [[circumcircle]] because the long diagonals all intersect each other at the polygon's center.

Special cases include:

A [[square]] has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is <math>\sqrt{2}\approx 1.414.</math>

A [[regular pentagon]] has five diagonals all of the same length. The ratio of a diagonal to a side is the [[golden ratio]], <math>\frac{1+\sqrt{5}}{2}\approx 1.618.</math>

A regular [[hexagon]] has nine diagonals: the six shorter ones are equal to each other in length; the three longer ones are equal to each other in length and intersect each other at the center of the hexagon. The ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is  <math>\sqrt{3}</math>.

A regular [[heptagon]] has 14 diagonals. The seven shorter ones equal each other, and the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal.

==Polyhedrons==
{{See also|Face diagonal|Space diagonal}}

A [[polyhedron]] (a [[solid object]] in [[three-dimensional space]], bounded by [[Plane (mathematics)|two-dimensional]] [[face (geometry)|faces]]) may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face; and space diagonals, entirely in the interior of the polyhedron (except for the endpoints on the vertices).

==Higher dimensions==
===N-Cube===
The lengths of an n-dimensional [[hypercube]]'s diagonals can be calculated by [[mathematical induction]]. The longest diagonal of an n-cube is <math>\sqrt{n}</math>. Additionally, there are <math>2^{n-1}\binom{n}{x+1}</math> of the ''xth'' shortest diagonal. As an example, a 5-cube would have the diagonals:

{|rules="none" border="0" cellspacing="4" cellpadding="0" style="background:transparent;text-align:right"
|-valign="top"
|
{|class="wikitable"
|-
! Diagonal length !! Number of diagonals
|-
| {{sqrt|2}} || 160
|-
| {{sqrt|3}} || 160
|-
| 2 || 80
|-
| {{sqrt|5}} || 16
|}
|}

Its total number of diagonals is 416. In general, an n-cube has a total of <math>2^{n-1}(2^n-n-1)</math> diagonals. This follows from the more general form of <math>\frac{v(v-1)}{2}-e</math> which describes the total number of face and space diagonals in convex [[polytope]]s.<ref>{{cite web | url=https://www.themathdoctors.org/counting-diagonals-of-a-polyhedron/ | title=Counting Diagonals of a Polyhedron – the Math Doctors }}</ref> Here, v represents the number of vertices and e represents the number of edges.

==Geometry==
By analogy, the [[subset]] of the [[Cartesian product]] ''X''&times;''X'' of any set ''X'' with itself, consisting of all pairs {{tmath|(x,x)}}, is called the diagonal, and is the [[Graph of a relation|graph]] of the [[Equality (mathematics)|equality]] [[Relation (mathematics)|relation]] on ''X'' or equivalently the [[Graph of a function|graph]] of the [[identity function]] from ''X'' to ''X''.  This plays an important part in geometry; for example, the [[fixed point (mathematics)|fixed point]]s of a [[function (mathematics)|mapping]] ''F'' from ''X'' to itself may be obtained by intersecting the graph of ''F'' with the diagonal.

In geometric studies, the idea of intersecting the diagonal ''with itself'' is common, not directly, but by perturbing it within an [[equivalence class]]. This is related at a deep level with the [[Euler characteristic]] and the zeros of [[vector field]]s. For example, the [[circle]] ''S''<sup>1</sup> has [[Betti number]]s 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-[[torus]] ''S''<sup>1</sup>&times;S<sup>1</sup> and observe that it can move ''off itself'' by the small motion (''θ'', ''θ'') to (''θ'', ''θ'' + ''ε'').  In general, the intersection number of the graph of a function with the diagonal may be computed using [[Homology (mathematics)|homology]] via the [[Lefschetz fixed-point theorem]]; the self-intersection of the diagonal is the special case of the identity function.

==Notes==
{{reflist}}

==External links==
{{Wiktionary|diagonal}}
*[http://www.mathopenref.com/polygondiagonal.html Diagonals of a polygon] with interactive animation
*[http://mathworld.wolfram.com/PolygonDiagonal.html Polygon diagonal] from [[MathWorld]].

[[Category:Elementary geometry]]