In geometry a polygon (IPA: /ˈpɒlɨɡɒn, ˈpɒliɡɒn/) is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points e. , by a closed polygonal chain). A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of Line segments More formally a polygonal These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations
Usually two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge.
The basic geometrical notion has been adapted in various ways to suit particular purposes. For example in the computer graphics (image generation) field, the term polygon has taken on a slightly altered meaning, more related to the way the shape is stored and manipulated within the computer. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit
Polygons are primarily classified by the number of sides, see naming polygons below. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit
Polygons may be characterised by their degree of convexity:
We will assume Euclidean geometry throughout. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.
Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are:
The exterior angle is the supplementary angle to the interior angle. A pair of Angles is supplementary if their measurements add up to 180 degrees If the two supplementary angles are adjacent (i From this the sum of the interior angles can be easily confirmed, even if some interior angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between -½ and ½ winding. The term winding number may also refer to the Rotation number of an Iterated map. )
The area of a polygon is the measurement of the 2-dimensional region enclosed by the polygon. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. For a non-self-intersecting (simple) polygon with n vertices, the area and centroid are given by:
To close the polygon, the first and last vertices are the same, ie xn,yn = x0,y0. In Geometry, a simple polygon is a polygon whose sides do not intersect Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes The vertices must be ordered clockwise or counterclockwise, if they are ordered clockwise the area will be negative but correct in absolute value. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
The formula was described by Meister in 1769 and by Gauss in 1795. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral
The area A of a simple polygon can also be computed if the lengths of the sides, a1,a2, . In Geometry, a simple polygon is a polygon whose sides do not intersect . . , an and the exterior angles, θ1,θ2, . Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle . . , θn are known. The formula is
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 a simple formula for the polygon's area based on the numbers of interior and boundary grid points. Given a Simple polygon constructed on a grid of equal-distanced points (i
If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem. In Geometry, the Bolyai–Gerwien theorem states that if two simple Polygons of equal Area are given one can cut the first into finitely many polygonal
For a regular polygon with n sides of length s, the area is given by:
The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer:
An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape. For information on degrees of freedom in other sciences see Degrees of freedom. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined In the case of a line of symmetry the latter reduces to n-2. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect
Let k≥2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n-1 degrees of freedom.
In a broad sense, a polygon is an unbounded sequence or circuit of alternating segments (sides) and angles (corners). The modern mathematical understanding is to describe this structural sequence in terms of an 'abstract' polygon which is a partially-ordered set (poset) of elements. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.
A geometric polygon is understood to be a 'realization' of the associated abstract polygon; this involves some 'mapping' of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations As another example, most polygons are unbounded because they close back on themselves, while apeirogons (infinite polygons) are unbounded because they go on for ever so you can never reach any bounding end point. So when we talk about "polygons" we must be careful to explain what kind we are talking about.
A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. Add another arc of a different great circle and you have a digon. Tile the sphere with digons and you have a polyhedron called a hosohedron. Take just one great circle instead, run it all the way round, and add just one "corner" point, and you have a monogon or henagon.
Other realizations of these polygons are possible on other surfaces - but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate. for the degeneracy of a Graph, see Arboricity#Related_concepts.
The idea of a polygon has been generalised in various ways. Here is a short list of some degenerate cases (or special cases, depending on your point of view):
The word 'polygon' comes from Late Latin polygōnum (a noun), from Greek polygōnon/polugōnon πολύγωνον, noun use of neuter of polygōnos/polugōnos πολύγωνος (the masculine adjective), meaning "many-angled". Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Numerical prefixes are usually derived from the words for numbers in various languages most commonly Greek and Latin, although this is not always the case g. pentagon, dodecagon. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Regular dodecagon It usually refers to a regular dodecagon having all sides of equal length and all angles equal to 150° The triangle, quadrilateral, and nonagon are exceptions. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. Graphs and stars The K9 Complete graph is often drawn as a regular nonagon with all 36 edges connected For large numbers, mathematicians usually write the numeral itself, e. A mathematician is a person whose primary area of study and research is the field of Mathematics. g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information
Some special polygons also have their own names; for example, the regular star pentagon is also known as the pentagram. Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC
|henagon (or monogon)||1|
|triangle (or trigon)||3|
|quadrilateral (or tetragon)||4|
|heptagon (avoid "septagon" = Latin [sept-] + Greek)||7|
|enneagon (or nonagon)||9|
|hendecagon (avoid "undecagon" = Latin [un-] + Greek)||11|
|dodecagon (avoid "duodecagon" = Latin [duo-] + Greek)||12|
|tridecagon (or triskaidecagon)||13|
|tetradecagon (or tetrakaidecagon)||14|
|pentadecagon (or quindecagon or pentakaidecagon)||15|
|hexadecagon (or hexakaidecagon)||16|
|heptadecagon (or heptakaidecagon)||17|
|octadecagon (or octakaidecagon)||18|
|enneadecagon (or enneakaidecagon or nonadecagon)||19|
|No established English name|
"hectogon" is the Greek name (see hectometre),
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows
The 'kai' is not always used. A googol is the Large number 10100 that is the digit 1 followed by one hundred zeros (in Decimal representation Opinions differ on exactly when it should, or need not, be used (see also examples above).
That is, a 42-sided figure would be named as follows:
|Tens||and||Ones||final suffix||full polygon name|
and a 50-sided figure
|Tens||and||Ones||final suffix||full polygon name|
But beyond enneagons and decagons, professional mathematicians generally prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). MathWorld is an online Mathematics reference work created and largely written by Eric W Exceptions exist for side numbers that are difficult to express in numerical form.
Numerous regular polygons may be seen in nature. The Giant's Causeway (or Clochán na bhFómharach is an area of about 40000 interlocking Basalt columns the result of an ancient volcanic eruption In the world of minerals, crystals often have faces which are triangular, square or hexagonal. Quasicrystals can even have regular pentagons as faces. Quasicrystals are structural forms that are both ordered and nonperiodic Another fascinating example of regular polygons occurs when the cooling of lava forms areas of tightly packed hexagonal columns of basalt, which may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California. Lava is molten rock expelled by a Volcano during an eruption When first expelled from a volcanic vent it is a Liquid at Temperatures Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees Basalt (bəˈsɔːlt ˈbeisɔːlt ˈbæsɔːlt is a common Extrusive Volcanic rock. The Giant's Causeway (or Clochán na bhFómharach is an area of about 40000 interlocking Basalt columns the result of an ancient volcanic eruption Ireland (pronounced /ˈaɾlənd/ Éire) is the third largest island in Europe, and the twentieth-largest island in the world Devils Postpile is a dark cliff of Columnar basalt near Mammoth Mountain in California ( is a US state on the West Coast of the United States, along the Pacific Ocean.
The most famous hexagons in nature are found in the animal kingdom. ||-||-|}The carambola is a species of Tree native to Indonesia, India and Sri Lanka and is popular throughout Southeast Asia, The wax honeycomb made by bees is an array of hexagons used to store honey and pollen, and as a secure place for the larvae to grow. A honeycomb is a mass of Hexagonal Wax cells built by Honey bees in their nests to contain their larvae and stores of Honey and Bees are flying Insects closely related to Wasps and Ants Bees are a Monophyletic lineage within the superfamily Apoidea Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees There also exist animals who themselves take the approximate form of regular polygons, or at least have the same symmetry. For example, starfish display the symmetry of a pentagon or, less frequently, the heptagon or other polygons. Starfish (also called sea stars) are any Echinoderms belonging to the class Asteroidea. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass Other echinoderms, such as sea urchins, sometimes display similar symmetries. Echinoderms (Phylum Echinodermata) are a phylum of marine Animals (including Sea stars) Sea urchins are small globular spiny sea cat animals composing most of class Echinoidea. Though echinoderms do not exhibit exact radial symmetry, jellyfish and comb jellies do, usually fourfold or eightfold. "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. Jellyfish are free-swimming members of the phylum Cnidaria. They have several different basic morphologies that represent several different cnidarian classes including the The phylum Ctenophora (tɨˈnɒfərə commonly known as comb jellies, is a phylum that includes the Sea gooseberry ( Pleurobrachia pileus) and
Radial symmetry (and other symmetry) is also widely observed in the plant kingdom, particularly amongst flowers, and (to a lesser extent) seeds and fruit, the most common form of such symmetry being pentagonal. A particularly striking example is the Starfruit, a slightly tangy fruit popular in Southeast Asia, whose cross-section is shaped like a pentagonal star. ||-||-|}The carambola is a species of Tree native to Indonesia, India and Sri Lanka and is popular throughout Southeast Asia,
Moving off the earth into space, early mathematicians doing calculations using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are orbiting one another, there exist certain points in space, called Lagrangian points, where a smaller body (such as an asteroid or a space station) will remain in a stable orbit. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The sun-earth system has five Lagrangian points. The two most stable are exactly 60 degrees ahead and behind the earth in its orbit; that is, joining the centre of the sun and the earth and one of these stable Lagrangian points forms an equilateral triangle. Astronomers have already found asteroids at these points. It is still debated whether it is practical to keep a space station at the Lagrangian point — although it would never need course corrections, it would have to frequently dodge the asteroids that are already present there. There are already satellites and space observatories at the less stable Lagrangian points.
A polygon in a computer graphics (image generation) system is a two-dimensional shape that is modelled and stored within its database. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data A polygon can be coloured, shaded and textured, and its position in the database is defined by the co-ordinates of its vertices (corners).
Naming conventions differ from those of mathematicians:
Use of Polygons in Real-time imagery. 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 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 so that as the viewing point moves through the scene, it is perceived in 3D.
Morphing. To avoid artificial effects at polygon boundaries where the planes of contiguous polygons are at different angle, so called 'Morphing Algorithms' are used. These blend, soften or smooth the polygon edges so that the scene looks less artificial and more like the real world.
Polygon Count. Since a polygon can have many sides and need many points to define it, in order to compare one imaging system with another, "polygon count" is generally taken as a triangle. A triangle is processed as three points in the x,y, and z axes, needing nine geometrical descriptors. In addition, coding is applied to each polygon for colour, brightness, shading, texture, NVG (intensifier or night vision), Infra-Red characteristics and so on. When analysing the characteristics of a particular imaging system, the exact definition of polygon count should be obtained as it applies to that system.
Meshed Polygons. The number of meshed polygons (`meshed' is like a fish net) can be up to twice that of free-standing unmeshed polygons, particularly if the polygons are contiguous. If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. There are (n+1) 2/2n2 vertices per triangle. Where n is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).
Vertex Count. Because of effects such as the above, a count of Vertices may be more reliable than Polygon count as an indicator of the capability of an imaging system.
Point in polygon test. In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line segments. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. It is known as the Point in polygon test. In Computational geometry, the point-in-polygon ( PIP) problem asks whether a given point in the plane lies inside outside or on the boundary of a Polygon