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Octagon In geometry , an OCTAGON (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eightsided polygon or 8gon. A regular octagon has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square , t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon , t{16} [...More...]  "Octagon" on: Wikipedia Yahoo 

Circumscribed Circle In geometry , the CIRCUMSCRIBED CIRCLE or CIRCUMCIRCLE of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the CIRCUMCENTER and its radius is called the CIRCUMRADIUS. A polygon which has a circumscribed circle is called a CYCLIC POLYGON (sometimes a CONCYCLIC POLYGON, because the vertices are concyclic ). All regular simple polygons , all isosceles trapezoids , all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle , which is the smallest circle that completely contains the polygon within it. Not every polygon has a circumscribed circle, as the vertices of a polygon do not need to all lie on a circle, but every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm [...More...]  "Circumscribed Circle" on: Wikipedia Yahoo 

Apothem The APOTHEM (sometimes abbreviated as APO ) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent . For a regular pyramid , which is a pyramid whose base is a regular polygon, the apothem is the slant height of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a plane parallel to the base), the apothem is the height of a trapezoidal lateral face [...More...]  "Apothem" on: Wikipedia Yahoo 

Inscribed Figure In geometry , an INSCRIBED planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron ) is tangent to every side or face of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure [...More...]  "Inscribed Figure" on: Wikipedia Yahoo 

Central Angle Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians ). The central angle is also known as the arc's angular distance . The size of a central angle Θ is 0° < Θ < 360° or 0 < Θ < 2π (radians). When defining or drawing a central angle, in addition to specifying the points A and B, one must specify whether the angle being defined is the convex angle (180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise. CONTENTS * 1 Formulas * 2 Central angle Central angle of a regular polygon * 3 See also * 4 References * 5 External links FORMULASIf the intersection points A and B of the legs of the angle with the circle form a diameter , then Θ = 180° is a straight angle . (In radians, Θ = π.) Let L be the MINOR ARC of the circle between points A and B, and let R be the radius of the circle. Central angle. Convex. Is subtended by minor arc LIf the central angle Θ is subtended by L, then 0 , = ( 180 L R ) = L R . {displaystyle 0^{circ } 2 R = ( 180 L R ) [...More...]  "Central Angle" on: Wikipedia Yahoo 

Radian The RADIAN is the standard unit of angular measure, used in many areas of mathematics . The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at A072097 ). The unit was formerly an SI supplementary unit , but this category was abolished in 1995 and the radian is now considered an SI derived unit SI derived unit . Separately, the SI unit of solid angle measurement is the steradian . The radian is represented by the symbol RAD. An alternative symbol is c, the superscript letter c, for "circular measure", or the letter r, but both of those symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r). So for example, a value of 1.2 radians could be written as 1.2 rad, 1.2 r, 1.2rad, or 1.2c [...More...]  "Radian" on: Wikipedia Yahoo 

Orthodiagonal Quadrilateral In Euclidean geometry Euclidean geometry , an ORTHODIAGONAL QUADRILATERAL is a quadrilateral in which the diagonals cross at right angles . In other words, it is a foursided figure in which the line segments between nonadjacent vertices are orthogonal (perpendicular) to each other. CONTENTS * 1 Special Special cases * 2 Characterizations * 2.1 Comparison with a tangential quadrilateral * 3 Area Area * 4 Other properties * 5 Properties of orthodiagonal quadrilaterals that are also cyclic * 5.1 Circumradius Circumradius and area * 5.2 Other properties * 6 References SPECIAL CASESA kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals [...More...]  "Orthodiagonal Quadrilateral" on: Wikipedia Yahoo 

Midpoint Polygon In geometry , the MIDPOINT POLYGON of a polygon P is the polygon whose vertices are the midpoints of the edges of P. It is sometimes called the KASNER POLYGON after Edward Kasner , who termed it the inscribed polygon "for brevity". The medial triangle The Varignon parallelogram CONTENTS* 1 Examples * 1.1 Triangle * 1.2 Quadrilateral * 2 See also * 3 References * 4 Further reading * 5 External links EXAMPLESTRIANGLEThe midpoint polygon of a triangle is called the medial triangle . It shares the same centroid and medians with the original triangle. The perimeter of the medial triangle equals the semiperimeter of the original triangle, and the area is one quarter of the area of the original triangle. This can be proven by the midpoint theorem of triangles and Heron\'s formula . The orthocenter of the medial triangle coincides with the circumcenter of the original triangle [...More...]  "Midpoint Polygon" on: Wikipedia Yahoo 

Angle 2D ANGLES * Right * Interior * Exterior 2D ANGLE PAIRS * Adjacent * Vertical * Complementary * Supplementary * Transversal 3D ANGLES * Dihedral An angle formed by two rays emanating from a vertex. ∠, the angle symbol in Unicode Unicode is U+2220 . In planar geometry , an ANGLE is the figure formed by two rays , called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane . Angles are also formed by the intersection of two planes in Euclidean and other spaces . These are called dihedral angles . Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection [...More...]  "Angle" on: Wikipedia Yahoo 

Coefficients In mathematics , a COEFFICIENT is a multiplicative factor in some term of a polynomial , a series or any expression ; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters , and must be clearly distinguished from the other variables. For example, in 7 x 2 3 x y + 1.5 + y , {displaystyle 7x^{2}3xy+1.5+y,} the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem or any expression in these parameters [...More...]  "Coefficients" on: Wikipedia Yahoo 

Pi The number π is a mathematical constant , the ratio of a circle 's circumference to its diameter , commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid18th century, though it is also sometimes spelled out as "PI" (/paɪ/ ). Being an irrational number , π cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern ). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed . In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness , but to date no proof of this has been discovered. Also, π is a transcendental number ; that is, a number that is not the root of any nonzero polynomial having rational coefficients [...More...]  "Pi" on: Wikipedia Yahoo 

Coxeter HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a Britishborn Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London London but spent most of his adult life in Canada Canada . He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. He worked for 60 years at the University of Toronto University of Toronto and published twelve books . He was most noted for his work on regular polytopes and higherdimensional geometries [...More...]  "Coxeter" on: Wikipedia Yahoo 

Petrie Polygon In geometry , a PETRIE POLYGON for a regular polytope of n dimensions is a skew polygon such that every (n1) consecutive side (but no n) belongs to one of the facets . The PETRIE POLYGON of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side (but no three) belongs to one of the faces . For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, h, is Coxeter number of the Coxeter group Coxeter group . These polygons and projected graphs are useful in visualizing symmetric structure of the higherdimensional regular polytopes [...More...]  "Petrie Polygon" on: Wikipedia Yahoo 

Orthogonal Projection In linear algebra and functional analysis , a PROJECTION is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent ). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection . One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object [...More...]  "Orthogonal Projection" on: Wikipedia Yahoo 

Compass (drawing Tool) A PAIR OF COMPASSES, also known simply as a COMPASS, is a technical drawing instrument that can be used for inscribing circles or arcs . As dividers , they can also be used as tools to measure distances, in particular on maps . Compasses can be used for mathematics , drafting , navigation and other purposes. Compasses are usually made of metal or plastic, and consist of two parts connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the other part a pencil, or sometimes a pen. Prior to computerization, compasses and other tools for manual drafting were often packaged as a "bow set" with interchangeable parts . Today these facilities are more often provided by computeraided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry , technical drawing, etc [...More...]  "Compass (drawing Tool)" on: Wikipedia Yahoo 

Straightedge A STRAIGHTEDGE is a tool with an edge free from curves, or straight, used for transcribing straight lines, or checking the straightness of lines. If it has equally spaced markings along its length, it is usually called a ruler . Straightedges are used in the automotive service and machining industry to check the flatness of machined mating surfaces. True straightness can in some cases be checked by using a laser line level as an optical straightedge: it can illuminate an accurately straight line on a flat surface such as the edge of a plank or shelf. A pair of straightedges called winding sticks are used in woodworking to amplify twist (wind) in pieces of wood [...More...]  "Straightedge" on: Wikipedia Yahoo 