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Triangle
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices _A_, _B_, and _C_ is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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Edge (geometry)
In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higher-dimensional polytope . In a polygon, an edge is a line segment on the boundary, and is often called a SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn graph theory , an edge is an abstract object connecting two graph vertices , unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place
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Schläfli Symbol
In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol
Schläfli symbol
is named after the 19th-century Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas
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Area
AREA is the quantity that expresses the extent of a two-dimensional figure or shape , or planar lamina , in the plane . Surface area
Surface area
is its analog on the two-dimensional surface of a three-dimensional object . Area
Area
can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units
International System of Units
(SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long
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Internal Angle
2D ANGLES * Right * Interior * Exterior-------------------------2D ANGLE PAIRS * Adjacent * Vertical * Complementary * Supplementary * Transversal -------------------------3D ANGLES * Dihedral Internal and External angles In geometry , an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex , this angle is called an INTERIOR ANGLE (or INTERNAL ANGLE) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex . If every internal angle of a simple polygon is less than 180°, the polygon is called convex . In contrast, an EXTERIOR ANGLE (or EXTERNAL ANGLE) is an angle formed by one side of a simple polygon and a line extended from an adjacent side . :pp
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Degree (angle)
A DEGREE (in full, a DEGREE OF ARC, ARC DEGREE, or ARCDEGREE), usually denoted by ° (the degree symbol ), is a measurement of a plane angle , defined so that a full rotation is 360 degrees. It is not an SI unit , as the SI unit of angular measure is the radian , but it is mentioned in the SI brochure as an accepted unit . Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians. CONTENTS * 1 History * 2 Subdivisions * 3 Alternative units * 4 See also * 5 Notes * 6 References * 7 External links HISTORY See also: Decans
Decans
A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle. The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year
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Polygon
In elementary geometry , a POLYGON (/ˈpɒlɪɡɒn/ ) is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or _circuit_. These segments are called its _edges _ or _sides_, and the points where two edges meet are the polygon's _vertices _ (singular: vertex) or _corners_. The interior of the polygon is sometimes called its _body_. An _N_-GON is a polygon with _n_ sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other self-intersecting polygons
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Shape
A SHAPE is the form of an object or its external boundary, outline, or external surface , as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons . Examples of geons include cones and spheres. CONTENTS * 1 Classification of simple shapes * 2 Shape in geometry * 2.1 Equivalence of shapes * 2.2 Congruence and similarity * 2.3 Homeomorphism * 3 Shape analysis * 4 Similarity classes * 5 See also * 6 References * 7 External links CLASSIFICATION OF SIMPLE SHAPES Main article: Lists of shapes A variety of polygonal shapes. Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles , quadrilaterals , pentagons , etc
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; _geo-_ "earth", _-metron_ "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer . Geometry arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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Euclidean Geometry
EUCLIDEAN GEOMETRY is a mathematical system attributed to the Alexandrian Greek mathematician Euclid , which he described in his textbook on geometry : the _Elements _. Euclid's method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions (theorems ) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system . The _Elements_ begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof . It goes on to the solid geometry of three dimensions . Much of the _Elements_ states results of what are now called algebra and number theory , explained in geometrical language
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Plane (mathematics)
In mathematics , a PLANE is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space . Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry . When working exclusively in two-dimensional Euclidean space , the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, geometry , trigonometry , graph theory , and graphing are performed in a two-dimensional space, or, in other words, in the plane
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Euclidean Space
In geometry , EUCLIDEAN SPACE encompasses the two-dimensional Euclidean plane , the three-dimensional space of Euclidean geometry , and certain other spaces. It is named after the Ancient Greek mathematician Euclid
Euclid
of Alexandria . The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry . Euclidean spaces also generalize to higher dimensions . Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates , while the other properties of these spaces were deduced as theorems . Geometric constructions are also used to define rational numbers . When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space
Euclidean space
using Cartesian coordinates
Cartesian coordinates
and the ideas of analytic geometry
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