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56-596: 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 n ( n − 3 ) 2 {\displaystyle {\tfrac {n(n-3)}{2}}} total diagonals, as each vertex has diagonals to all other vertices except itself and

112-419: A , b , c , d , e {\displaystyle a,b,c,d,e} and diagonals d 1 , d 2 , d 3 , d 4 , d 5 {\displaystyle d_{1},d_{2},d_{3},d_{4},d_{5}} , the following inequality holds: A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form

168-398: A polygon ( / ˈ p ɒ l ɪ ɡ ɒ n / ) is a plane figure made up of line segments connected to form a closed polygonal chain . The segments of a closed polygonal chain are called its edges or sides . The points where two edges meet are the polygon's vertices or corners . An n -gon is a polygon with n sides; for example, a triangle is a 3-gon. A simple polygon

224-470: A regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 3 1 ⁄ 3 (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons: The maximum known packing density of

280-494: A given perimeter, the one with the largest area is regular (and therefore cyclic). Many specialized formulas apply to the areas of regular polygons . The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by This radius is also termed its apothem and is often represented as a . The area of a regular n -gon in terms of the radius R of its circumscribed circle can be expressed trigonometrically as: The area of

336-465: A range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal). A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth

392-401: A regular n -gon inscribed in a unit-radius circle, with side s and interior angle α , {\displaystyle \alpha ,} can also be expressed trigonometrically as: The area of a self-intersecting polygon can be defined in two different ways, giving different answers: Using the same convention for vertex coordinates as in the previous section, the coordinates of

448-444: A regular pentagon is ( 5 − 5 ) / 3 ≈ 0.921 {\displaystyle (5-{\sqrt {5}})/3\approx 0.921} , achieved by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has

504-399: A simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1. In every polygon with perimeter p and area A , the isoperimetric inequality p 2 > 4 π A {\displaystyle p^{2}>4\pi A} holds. For any two simple polygons of equal area,

560-471: A simple polygon given by a sequence of line segments. This is called the point in polygon test. Regular pentagon In geometry , a pentagon (from Greek πέντε (pente)  'five' and γωνία (gonia)  'angle' ) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting . A self-intersecting regular pentagon (or star pentagon )

616-568: 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 This is OEIS sequence A006522. 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 ( n 4 ) {\displaystyle \textstyle {\binom {n}{4}}} . This holds, for example, for any regular polygon with an odd number of sides. The formula follows from

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672-463: Is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes. The word polygon derives from the Greek adjective πολύς ( polús ) 'much', 'many' and γωνία ( gōnía ) 'corner' or 'angle'. It has been suggested that γόνυ ( gónu ) 'knee' may be the origin of gon . Polygons are primarily classified by

728-492: Is called a pentagram . A regular pentagon has Schläfli symbol {5} and interior angles of 108°. A regular pentagon has five lines of reflectional symmetry , and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t , {\displaystyle t,} its height H {\displaystyle H} (distance from one side to

784-439: Is commonly called the shoelace formula or surveyor's formula . The area A of a simple polygon can also be computed if the lengths of the sides, a 1 , a 2 , ..., a n and the exterior angles , θ 1 , θ 2 , ..., θ n are known, from: The formula was described by Lopshits in 1963. If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives

840-464: Is concerned only with simple and solid polygons, a polygon may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygons and other self-intersecting polygons . Some sources also consider closed polygonal chains in Euclidean space to be a type of polygon (a skew polygon ), even when the chain does not lie in a single plane. A polygon

896-451: Is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines). The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of

952-429: Is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a solid polygon . The interior of a solid polygon is its body , also known as a polygonal region or polygonal area . In contexts where one

1008-447: Is the perimeter of the polygon, and r is the inradius (equivalently the apothem ). Substituting the regular pentagon's values for P and r gives the formula with side length t . Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle . The apothem , which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by Like every regular convex polygon,

1064-467: Is to look at the diagonal on the two- torus S xS 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 via the Lefschetz fixed-point theorem ; the self-intersection of the diagonal is the special case of the identity function. Polygon In geometry ,

1120-461: The Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides do determine the area. Of all n -gons with given side lengths, the one with the largest area is cyclic. Of all n -gons with

1176-635: The Giant's Causeway in Northern Ireland , or at the Devil's Postpile in California . In biology , the surface of the wax honeycomb made by bees is an array of hexagons , and the sides and base of each cell are also polygons. In computer graphics , a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of

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1232-564: The fixed points of a 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 fields . For example, the circle S has Betti numbers 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this

1288-426: The g5 subgroup has no degrees of freedom but can be seen as directed edges . A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio . An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take

1344-456: The geometrical vertices , as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials . Any surface is modelled as a tessellation called polygon mesh . If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2 n squared triangles since there are two triangles in a square. There are ( n + 1) / 2( n ) vertices per triangle. Where n

1400-488: The regular star pentagon is also known as the pentagram . To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by Kepler , and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra , though not all sources use it. Polygons have been known since ancient times. The regular polygons were known to

1456-517: The xth shortest diagonal. As an example, a 5-cube would have the diagonals: Its total number of diagonals is 416. In general, an n-cube has a total of 2 n − 1 ( 2 n − n − 1 ) {\displaystyle 2^{n-1}(2^{n}-n-1)} diagonals. This follows from the more general form of v ( v − 1 ) 2 − e {\displaystyle {\frac {v(v-1)}{2}}-e} which describes

1512-539: The ancient Greeks, with the pentagram , a non-convex regular polygon ( star polygon ), appearing as early as the 7th century B.C. on a krater by Aristophanes , found at Caere and now in the Capitoline Museum . The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century. In 1952, Geoffrey Colin Shephard generalized the idea of polygons to

1568-453: The centroid of a solid simple polygon are In these formulas, the signed value of area A {\displaystyle A} must be used. For triangles ( n = 3 ), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3 . The centroid of the vertex set of a polygon with n vertices has the coordinates The idea of a polygon has been generalized in various ways. Some of

1624-401: The circle at point P , and chord PD is the required side of the inscribed pentagon. To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as 5 / 2 {\displaystyle \scriptstyle {\sqrt {5}}/2} . Side h of

1680-461: The circumradius R {\displaystyle R} of a regular pentagon is given, its edge length t {\displaystyle t} is found by the expression and its area is since the area of the circumscribed circle is π R 2 , {\displaystyle \pi R^{2},} the regular pentagon fills approximately 0.7568 of its circumscribed circle. The area of any regular polygon is: where P

1736-426: The complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons . Polygons appear in rock formations, most commonly as the flat facets of crystals , where the angles between the sides depend on the type of mineral from which the crystal is made. Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt , which may be seen at

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1792-490: The construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D . Angle CMD is bisected, and the bisector intersects the vertical axis at point Q . A horizontal line through Q intersects

1848-454: The cosine double angle formula . This is the cosine of 72°, which equals ( 5 − 1 ) / 4 {\displaystyle \left({\sqrt {5}}-1\right)/4} as desired. The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation . This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows: Steps 6–8 are equivalent to

1904-505: The distances from the vertices of a regular pentagon to any point on its circumcircle, then The regular pentagon is constructible with compass and straightedge , as 5 is a Fermat prime . A variety of methods are known for constructing a regular pentagon. Some are discussed below. One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's Polyhedra . The top panel shows

1960-401: 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. 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 ,

2016-535: The following version, shown in the animation: A regular pentagon is constructible using a compass and straightedge , either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC. The regular pentagon has Dih 5 symmetry , order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih 1 , and 2 cyclic group symmetries: Z 5 , and Z 1 . These 4 symmetries can be seen in 4 distinct symmetries on

2072-462: The interior of the polyhedron (except for the endpoints on the vertices). The lengths of an n-dimensional hypercube 's diagonals can be calculated by mathematical induction . The longest diagonal of an n-cube is n {\displaystyle {\sqrt {n}}} . Additionally, there are 2 n − 1 ( n x + 1 ) {\displaystyle 2^{n-1}{\binom {n}{x+1}}} of

2128-444: 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

2184-833: The more important include: The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον ( polygōnon/polugōnon ), noun use of neuter of πολύγωνος ( polygōnos/polugōnos , the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek -derived numerical prefix with the suffix -gon , e.g. pentagon , dodecagon . The triangle , quadrilateral and nonagon are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon. Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example

2240-552: The notation ( x n , y n ) = ( x 0 , y 0 ) will also be used. If the polygon is non-self-intersecting (that is, simple ), the signed area is or, using determinants where Q i , j {\displaystyle Q_{i,j}} is the squared distance between ( x i , y i ) {\displaystyle (x_{i},y_{i})} and ( x j , y j ) . {\displaystyle (x_{j},y_{j}).} The signed area depends on

2296-525: The number of sides. Polygons may be characterized by their convexity or type of non-convexity: The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon . Euclidean geometry is assumed throughout. Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: In this section,

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2352-399: The opposite vertex), width W {\displaystyle W} (distance between two farthest separated points, which equals the diagonal length D {\displaystyle D} ) and circumradius R {\displaystyle R} are given by: The area of a convex regular pentagon with side length t {\displaystyle t} is given by If

2408-410: The optimal density among all packings of regular pentagons in the plane. There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around

2464-423: The ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive x -axis to the positive y -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value . This

2520-441: The pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126° . To find the number of sides this polygon has, the result is 360 / (180 − 126) = 6 2 ⁄ 3 , which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons. There are 15 classes of pentagons that can monohedrally tile

2576-534: The pentagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r10 and no symmetry is labeled a1 . The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal) or edges ( p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

2632-585: 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 2 ≈ 1.414. {\displaystyle {\sqrt {2}}\approx 1.414.} A regular pentagon has five diagonals all of the same length. The ratio of a diagonal to a side is the golden ratio , 1 + 5 2 ≈ 1.618. {\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.} A regular hexagon has nine diagonals:

2688-426: The processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation. In computer graphics and computational geometry , it is often necessary to determine whether a given point P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} lies inside

2744-627: The regular convex pentagon has a circumscribed circle . For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE. For an arbitrary point in the plane of a regular pentagon with circumradius R {\displaystyle R} , whose distances to the centroid of the regular pentagon and its five vertices are L {\displaystyle L} and d i {\displaystyle d_{i}} respectively, we have If d i {\displaystyle d_{i}} are

2800-399: 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. A polyhedron (a solid object in three-dimensional space , bounded by two-dimensional 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

2856-410: 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 3 {\displaystyle {\sqrt {3}}} . A regular heptagon has 14 diagonals. The seven shorter ones equal each other, and

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2912-444: The smaller triangle then is found using the half-angle formula : where cosine and sine of ϕ are known from the larger triangle. The result is: If DP is truly the side of a regular pentagon, m ∠ C D P = 54 ∘ {\displaystyle m\angle \mathrm {CDP} =54^{\circ }} , so DP = 2 cos(54°), QD = DP cos(54°) = 2cos (54°), and CQ = 1 − 2cos (54°), which equals −cos(108°) by

2968-441: The square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon. There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons . It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational. For all convex pentagons with sides

3024-534: The total number of face and space diagonals in convex polytopes. Here, v represents the number of vertices and e represents the number of edges. By analogy, the subset of the Cartesian product X × X of any set X with itself, consisting of all pairs (x,x), is called the diagonal, and is the graph of the equality relation on X or equivalently the graph of the identity function from X to X . This plays an important part in geometry; for example,

3080-451: The two adjacent vertices, or n  − 3 diagonals, and each diagonal is shared by two vertices. In general, a regular n -sided polygon has ⌊ n − 2 2 ⌋ {\displaystyle \lfloor {\frac {n-2}{2}}\rfloor } distinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square. In a convex polygon , if no three diagonals are concurrent at

3136-417: The vertices of the polygon under consideration are taken to be ( x 0 , y 0 ) , ( x 1 , y 1 ) , … , ( x n − 1 , y n − 1 ) {\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})} in order. For convenience in some formulas,

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