Diagonal
In geometry, a diagonal is a straight line segment that connects two nonconsecutive vertices of a polygon or polyhedron, excluding the sides or edges of the figure.[1] This concept is fundamental to understanding the internal structure of polygonal shapes, where diagonals divide the interior into triangles and facilitate calculations such as area determination for specific quadrilaterals like rhombi and kites.[2] For an n-sided polygon, the total number of diagonals is given by the formula \frac{n(n-3)}{2}, which accounts for the fact that each vertex connects to n-3 other non-adjacent vertices, avoiding double-counting.[3] In linear algebra, the term "diagonal" also describes a square matrix where all off-diagonal entries are zero, leaving only the main diagonal (from the top-left to bottom-right) potentially nonzero.[4] Diagonal matrices simplify computations, such as matrix powers and exponentiation, because multiplying them involves only scaling by their diagonal elements, and they play a central role in diagonalization processes that transform general matrices into easier-to-analyze forms.[5] Beyond geometry and algebra, diagonals appear in set theory through Cantor's diagonal argument, a proof by contradiction demonstrating the uncountability of the real numbers by constructing a number differing from each in an assumed countable list along its diagonal position.[6] This technique has broader applications in logic and computability, including Gödel's incompleteness theorems, highlighting the diagonal's role in foundational mathematics.[7]Basic Concepts
Definition
In geometry, a diagonal refers to a straight line segment that connects two non-adjacent vertices of a polygon or polyhedron.[8] For polygons in the plane, a diagonal joins vertices that do not share an edge, thereby lying entirely within the polygon's interior for convex cases.[3] This distinguishes diagonals from sides, which connect adjacent vertices.[9] In the context of polyhedra, diagonals are categorized into face diagonals and space diagonals. A face diagonal lies on one of the polyhedron's polygonal faces and connects non-adjacent vertices within that face, analogous to a two-dimensional diagonal.[10] In contrast, a space diagonal connects two vertices that do not share a common face, passing through the interior of the polyhedron.[11] These space diagonals highlight the three-dimensional structure, unlike face diagonals which remain planar.[12]Properties in Plane Geometry
In plane geometry, a diagonal is a line segment connecting two non-adjacent vertices of a polygon, lying entirely within the plane of the figure.[9] This distinguishes diagonals from sides, which connect adjacent vertices, and enables the decomposition of polygons into simpler triangular regions through triangulation. For instance, drawing all non-intersecting diagonals from one vertex divides an n-gon into n-2 triangles, facilitating proofs of properties like the sum of interior angles equaling (n-2)\pi radians.[9] In quadrilaterals, the simplest polygons with diagonals, each figure has exactly two diagonals that connect opposite vertices and generally intersect at a single interior point unless the quadrilateral is degenerate.[13] The properties of these diagonals vary by quadrilateral type. In a parallelogram, the diagonals bisect each other, meaning they intersect at their midpoints, a consequence of the vector equality of opposite sides.[14] Extending this, the diagonals of a rectangle are congruent in length and bisect each other, while in a rhombus, they are perpendicular and bisect the vertex angles. For a square, combining these traits, the diagonals are equal, perpendicular, and each bisects the angles at 45 degrees. Key theorems highlight the role of diagonals in relating geometric elements. Varignon's theorem states that connecting the midpoints of the sides of any quadrilateral forms a parallelogram, whose sides are parallel to the diagonals of the original quadrilateral and half their lengths; this Varignon parallelogram is a rhombus if the original diagonals are equal in length, and a rectangle if they are perpendicular.[15] For cyclic quadrilaterals inscribed in a circle, Ptolemy's theorem provides a multiplicative relation: the product of the lengths of the diagonals equals the sum of the products of the lengths of the opposite sides, expressed as AC \cdot BD = AB \cdot CD + AD \cdot BC, where AB, BC, CD, DA are the sides and AC, BD the diagonals.[16] This theorem, attributed to the ancient mathematician Ptolemy, underscores the interplay between diagonals and cyclic properties in plane figures.[17] Diagonals also influence angle and area measures. In a kite, one diagonal is the axis of symmetry and bisects the other, while both are perpendicular.[18] The area of a quadrilateral can be computed using the diagonals and the angle \theta between them via \frac{1}{2} d_1 d_2 \sin \theta, applicable to any convex quadrilateral where the diagonals intersect.[19] These properties extend conceptually to higher polygons, where diagonals may intersect to form additional vertices and edges, but in plane geometry, they fundamentally aid in classifying and analyzing polygonal structures.[9]In Polygons
Number of Diagonals
The number of diagonals in a convex polygon with n vertices (or sides) is given by the formula \frac{n(n-3)}{2}. This expression arises because the total number of line segments connecting any two vertices is the combination \binom{n}{2} = \frac{n(n-1)}{2}, which includes both the n sides of the polygon and the diagonals; subtracting the sides yields the diagonals: \frac{n(n-1)}{2} - n = \frac{n(n-3)}{2}.[20][21] This formula applies specifically to simple convex polygons, where no three diagonals intersect at a single interior point, ensuring all diagonals are distinct and lie within the polygon. For non-convex polygons, the count remains the same in terms of connections between vertices, but some diagonals may lie outside the boundary. The derivation assumes vertices are in convex position, as is standard in basic polygon geometry.[22][23] To illustrate, consider small polygons:| Polygon | n | Number of Diagonals |
|---|---|---|
| Triangle | 3 | 0 |
| Quadrilateral | 4 | 2 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 9 |
Regions and Intersections
In a convex polygon with n \geq 4 vertices, the diagonals intersect exclusively in the interior, provided the polygon is in general position where no three diagonals meet at a single interior point. Each such intersection arises from the crossing of two diagonals that connect four distinct vertices, forming a quadrilateral whose diagonals intersect at one point. Consequently, the total number of interior intersection points is \dbinom{n}{4}, as every set of four vertices determines exactly one such crossing.[8] These intersection points, combined with the polygon's n vertices and its \frac{n(n-3)}{2} diagonals, subdivide the interior into multiple regions. Applying Euler's formula for planar graphs (V - E + F = 2, where V is vertices including intersections, E is edges including diagonal segments, and F is faces including the exterior), or through direct combinatorial enumeration, the number of interior regions r(n) is given by r(n) = \dbinom{n}{4} + \dbinom{n-1}{2}. This can also be expressed in expanded form as r(n) = \frac{n^4 - 6n^3 + 23n^2 - 42n + 24}{24}. The formula accounts for the initial polygonal face, the additions from diagonals and their segments, and the splits at intersections.[24][25] For small values of n, the results illustrate the growth: a quadrilateral (n=4) has \dbinom{4}{4} = [1](/page/1) intersection and r(4) = 4 regions; a pentagon (n=5) has 5 intersections and r(5) = 11 regions; a hexagon (n=6) has 15 intersections and r(6) = 25 regions. These configurations maximize the number of regions under the no-three-concurrent assumption, emphasizing the combinatorial complexity introduced by diagonal crossings.[8] If the polygon is regular or otherwise positioned such that multiple diagonals concur at interior points, the number of intersections decreases and the regions are fewer than in the general case, as shared points reduce the splitting effect. For example, in a regular hexagon, three long diagonals intersect at the center, yielding 24 regions rather than 25, though adjusted counts depend on the specific concurrencies. Such cases highlight the role of geometric constraints in altering the topological division.[8][26]Diagonals in Regular Polygons
In a regular n-gon inscribed in a circle of circumradius R, the diagonals connect non-adjacent vertices and subtend central angles that are multiples of $2\pi/n. The length of a diagonal spanning k vertices (where $2 \leq k \leq \lfloor n/2 \rfloor) is d_k = 2R \sin\left(\frac{k\pi}{n}\right).[27] This formula arises from the chord length in a circle, where the central angle is \theta = 2\pi k / n, and the chord length is $2R \sin(\theta/2). If expressed in terms of the side length s = 2R \sin(\pi/n), the diagonal length becomes d_k = s \frac{\sin(k\pi/n)}{\sin(\pi/n)}.[27] Due to rotational symmetry, there are \lfloor n/2 \rfloor - 1 distinct diagonal lengths, corresponding to the unique values of k excluding sides and diameters (when n is even).[28] A prominent example occurs in the regular pentagon (n=5), where all diagonals are equal in length and form a pentagram when connected. For a side length of 1, each diagonal has length \phi = \frac{[1](/page/1) + \sqrt{5}}{2}, the golden ratio, satisfying \phi^2 = \phi + [1](/page/1).[29] This ratio emerges from solving the geometry of intersecting diagonals, as the diagonal d satisfies d = [1](/page/1) + 2 \cdot \frac{[1](/page/1)}{d} by considering isosceles triangles formed within the pentagon.[30] In the regular hexagon (n=6), there are two distinct diagonal lengths: the shorter ones (spanning two vertices) measure s\sqrt{3}, and the longer ones (spanning three vertices) are diameters of length $2s, all intersecting at the center.[31] The intersections of diagonals in a regular n-gon exhibit rotational and reflectional symmetry, with each interior crossing typically involving exactly two diagonals, except in cases of higher concurrency (e.g., at the center for even n). Every set of four vertices determines a unique intersection point inside the polygon, but symmetries reduce the total count. The number of such interior intersection points is given by the formula I(n) = \binom{n}{4} + \frac{n}{24} \left( \binom{m}{2} (m^2 + 5) - 3(-1)^m \right), where m = n \mod 24, adjusted as a piecewise polynomial based on n modulo 24; this counts points with multiplicity for concurrent diagonals.[32] In the regular pentagon, each pair of diagonals intersects at a single point, dividing the polygon into 11 regions, with each intersection segment ratio equal to the golden ratio \phi : 1 : \phi.[30] For larger n, such as the heptagon, intersections form complex symmetric patterns without triple points except at vertices.[33]In Polyhedra
Face Diagonals
In geometry, a face diagonal of a polyhedron is a line segment connecting two non-adjacent vertices that lie on the same polygonal face, distinguishing it from edges and space diagonals that pass through the interior.[34] Unlike space diagonals, which connect vertices on different faces, face diagonals remain entirely on the surface of the polyhedron.[35] The total number of face diagonals in a polyhedron is determined by summing the diagonals across all its faces, where each face, being an n-gon, contributes \frac{n(n-3)}{2} diagonals.[35] For example, a cube with six square faces (n=4) has $6 \times \frac{4(4-3)}{2} = 12 face diagonals, with two per face.[36] In a square pyramid, the square base contributes 2 diagonals, while the four triangular faces (n=3) contribute none, yielding a total of 2 face diagonals.[36] This approach highlights how face diagonals depend on the polyhedron's facial structure rather than its overall volume. Face diagonals play a key role in measuring surface distances and verifying polyhedral properties, such as in rectangular prisms where the lengths are \sqrt{l^2 + w^2}, \sqrt{l^2 + h^2}, and \sqrt{w^2 + h^2} for dimensions l, w, and h, with each type appearing on two opposite faces.[37] In regular polyhedra like the dodecahedron, face diagonals on pentagonal faces follow the golden ratio, \phi = \frac{1 + \sqrt{5}}{2}, scaled by the edge length, underscoring their geometric harmony.[38]Space Diagonals
In a polyhedron, a space diagonal (also known as a body diagonal or interior diagonal) is a line segment that connects two vertices not lying on the same face, passing through the interior of the polyhedron. Unlike face diagonals, which lie entirely on one of the polyhedron's faces, space diagonals link vertices from distinct faces and do not coincide with any edges. This distinguishes them from the edges of the polyhedron, which connect adjacent vertices. The tetrahedron is the only convex polyhedron without space diagonals, as all pairs of vertices are either connected by an edge or lie on the same triangular face.[12] The number of space diagonals in a polyhedron can be calculated by first determining the total number of line segments connecting any two vertices, which is the combination \binom{v}{2} = \frac{v(v-1)}{2} where v is the number of vertices, then subtracting the number of edges e and the total number of face diagonals. The number of face diagonals is the sum over all faces of \frac{n_f(n_f - 3)}{2}, where n_f is the number of sides of the f-th face. Thus, the formula for the number of space diagonals d_s is: d_s = \frac{v(v-1)}{2} - e - \sum_f \frac{n_f(n_f - 3)}{2}. This approach accounts for all possible connections while excluding surface elements. Using Euler's formula v - e + f = 2 for convex polyhedra, the total number of diagonals (face plus space) simplifies to \frac{v(v-3)}{2} + 2 - f, but space diagonals require the additional subtraction of face diagonals.[36] For the Platonic solids, the numbers of space diagonals vary due to their symmetry and face structures, as shown in the table below. All faces are regular polygons, and triangular faces contribute no diagonals.| Platonic Solid | Vertices (v) | Edges (e) | Faces (f) | Face Diagonals | Space Diagonals |
|---|---|---|---|---|---|
| Tetrahedron | 4 | 6 | 4 (triangles) | 0 | 0 |
| Cube | 8 | 12 | 6 (squares) | 12 | 4 |
| Octahedron | 6 | 12 | 8 (triangles) | 0 | 3 |
| Dodecahedron | 20 | 30 | 12 (pentagons) | 60 | 100 |
| Icosahedron | 12 | 30 | 20 (triangles) | 0 | 36 |