In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" (from διά- dia-, "through", "across" and γωνία gonia, "angle", related to gony "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus ("slanting line").
In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner.
There are also many other non-mathematical uses.
Non-mathematical uses
In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not connected to the corners of the rectangle.
Diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name.
A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle.
In association football, the diagonal system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch.
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 polygons, some diagonals are outside of the polygon.
Any n-sided polygon (n ≥ 3), convex or concave, has total diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or n − 3 diagonals, and each diagonal is shared by two vertices.
In general, a regular n-sided polygon has distinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.
Regions formed by diagonals
In a convex polygon, if no three diagonals are concurrent at a single point in the interior, the number of regions that the diagonals divide the interior into is given by
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.
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 . 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
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:
This formula shows that as the number of sides approaches infinity, the xth shortest diagonal approaches the length (x+1)a. Additionally, the formula for the shortest diagonal simplifies in the case of x = 1:
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
A regular pentagon has five diagonals all of the same length. The ratio of a diagonal to a side is the golden ratio,
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 .
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.