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Octagon
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol
Schläfli symbol
8 [1] 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 .Contents1 Properties of the general octagon 2 Regular octagon2.1 Area 2.2 Circumradius
Circumradius
and inradius 2.3 Diagonals 2.4 Construction and elementary properties 2.5 Standard coordinates 2.6 Dissection3 Skew octagon3.1 Petrie polygons4 Symmetry 5 Uses of octagons5.1 Other uses6 Derived figures6.1 Related polytopes7 See also 8 References 9 External linksProperties of the general octagon[edit]The diagonals of the green quadrilateral are equal in length and at right angles to each otherThe sum of all the internal angles of any octagon is 1080°
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Reflective Symmetry
In mathematics, a reflection (also spelled reflexion)[1] is a mapping from a Euclidean space
Euclidean space
to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term reflection is sometimes used for a larger class of mappings from a Euclidean space
Euclidean space
to itself, namely the non-identity isometries that are involutions
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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
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On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences
On-Line Encyclopedia of Integer Sequences
(OEIS), also cited simply as Sloane's, is an online database of integer sequences. It was created and maintained by Neil Sloane
Neil Sloane
while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009.[3] Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, and is widely cited
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Angle
2D anglesRight Interior Exterior2D angle pairsAdjacent Vertical Complementary Supplementary Transversal3D anglesDihedralAn angle formed by two rays emanating from a vertex.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.[1] 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. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle
Angle
is also used to designate the measure of an angle or of a rotation
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Radian
The radian (SI symbol rad) is the SI unit for measuring angles, and 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.[1] Separately, the SI unit of solid angle measurement is the steradian. The radian is most commonly represented by the symbol rad.[2] An alternative symbol is c, the superscript letter c (for "circular measure"), the letter r, or a superscript R,[3] but these symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r)
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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).[1] 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°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise.Contents1 Formulas 2 Central angle
Central angle
of a regular polygon 3 See also 4 References 5 External linksFormulas[edit] If 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
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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
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Apothem
The apothem (sometimes abbreviated as apo[1]) 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
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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
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
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Pi
The number π (/paɪ/) is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes’ constant.[1] Being an irrational number, π cannot be expressed exactly as a common 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
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Equidiagonal Quadrilateral
In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length
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Unit Circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere. If (x, y) is a point on the unit circle's circumference, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 = 1. displaystyle x^ 2 +y^ 2 =1
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Circumradius
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
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Inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.[1] An excircle or escribed circle[2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.[3] The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors.[3][4] The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two
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Straightedge
A straightedge or straight edge[1] is a tool with a straight edge, used for drawing straight lines, or checking their straightness. 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.Contents1 Compass-and-straightedge construction 2 See also 3 References 4 External linksCompass-and-straightedge construction[edit] Main article: Compass and straightedge An idealized straightedge is used in compass-and-straightedge constructions in plane geometry
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