Image Apothem of hexagon.svg thumb right Apothem of a hexagon The apothem 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 congruence geometry congruent and have the same length. For a regular pyramid geometry 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 geometry plane parallel to the base , the apothem is the height of a trapezoidal lateral face. http www.bymath.com studyguide geo sec geo15.htm For a triangle necessarily equilateral , the apothem .... Properties of apothems The apothem a can be used to find the area of any regular n ... to the apothem multiplied by half the perimeter since ns p . math A frac nsa 2 frac pa 2 . math ... triangle Types of triangles isosceles triangles , and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height. An apothem ... 2 pi r r 2 pi r 2 math Finding the apothem The apothem of a regular polygon can be found multiple ways, of which two are described here. The apothem a of a regular n sided polygon with side length s , or circumradius ... n . math The apothem can also be found by math a frac 1 2 s tan left frac 90 circ n 2 n right . math ... because math s frac p n . math External links http www.mathopenref.com polygonapothem.html Apothem of a regular polygon With interactive animation http www.bymath.com studyguide geo sec geo15.htm Apothem of pyramid or truncated pyramid http demonstrations.wolfram.com SagittaApothemAndChord Sagitta, Apothem ... more details
. If you know it, please add it here. Well it s just 2 times apothem for even n, or apothem radius for odd ... from any interior point to the n sides is n times the apothem the apothem being the distance ... length, s or apothem , a math r frac s 2 sin frac pi n frac a cos frac pi n math Area This section ... with n 5 pentagon with side geometry side s , circumradius r and apothem a The area A of a convex regular n sided polygon having side geometry side s , circumradius r , apothem a , and perimeter p is given ... tfrac 2 pi n math For regular polygons with side s 1, resp. circumradius r 1, resp. apothem a 1, this produces ... 3 Area when circumradius r 1 style background 8080FF colspan 3 Area when apothem a 1 Exact Approximate ... more details
otheruses Image CIRCLE 1.svg thumb right Circle illustration In classical geometry , a radius of a circle or sphere is any line segment from its Centre geometry center to its perimeter . By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter . ref name mwd1 http www.mathwords.com r radius of a circle or sphere.htm Definition of radius at mathwords.com. Accessed on 2009 08 08. ref If the object does not have an obvious center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere . In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity. For regular polygons, the radius is the same as its circumradius. ref name schaum Barnett Rich, Christopher Thomas 2008 , Schaum s Outline of Geometry , 4th edition, 326 pages. McGraw Hill Professional. ISBN 0071544127, 9780071544122. http books.google.com.br books?id ab8lZG2yubcC Online version accessed on 2009 08 08. ref The inradius of a regular polygon is also called apothem . In graph theory , the radius graph theory radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph. ref name yel Jonathan L. Gross, Jay Yellen 2006 , Graph theory and its applications . 2nd edition, 779 pages CRC Press. ISBN 158488505X, 9781584885054. http books.google.com.br books?id unEloQ sYmkC Online version accessed on 2009 08 08. ref The name comes from Latin radius , meaning ray but also the spoke of a chariot wheel. ref name radic http dictionary.reference.com browse Radius Definition of Radius at dictionary.reference.com. Accessed on 2009 08 08. ref The plural of radius can be either radii or the conventional ... more details
Pi box The area of a disk the region inside a circle , often incorrectly called the area of a circle , is r sup 2 sup when the circle has radius r . Here the symbol Greek alphabet Greek letter Pi letter pi denotes, as usual, the constant ratio of the circumference of a circle to its diameter . It is easy to deduce the area of a disk mathematics disk from basic principles the area of a regular polygon is half its apothem times its perimeter, and a regular polygon becomes a circle as the number of sides increases, so the area of a disk is half its radius times its circumference i.e. frac 1 2 r 2 r . Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis . However, in Ancient Greece the great mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle s circumference and whose height equals the circle s radius in his book Measurement of a Circle . The circumference is 2 r , and the area of a triangle is half the base times the height, yielding the area r sup 2 sup for the disk. Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , ref name heath citation first Thomas L. last Heath authorlink Thomas Little Heath title A Manual of Greek Mathematics publisher Courier Dover Publications year 2003 isbn 0486432319 pages 121 132 url http books.google.com books?id HZNr mGFzQC&pg PA121 . ref but did not identify the constant of proportionality . Using polygons The area of a regular polygon is half its perimeter times the apothem . As the number of sides of the regular polygon increases, it becomes identical to a circle, and the apothem becomes identical to the radius. Therefore, the area of a circle is half its circumference times the radius. ref Hi ... more details
Refimprove date July 2008 Odd polygon db Odd polygon stat table p5 wiktionarypar pentagon two other uses the geometric figure the headquarters of the United States Department of Defense The Pentagon In geometry , a pentagon from pente , which is Greek language Greek for the number 5 is any five sided polygon . A pentagon may be simple or self intersecting. The sum of the internal angle s in a simple polygon simple pentagon is 540 . A pentagram is an example of a self intersecting pentagon. Regular pentagons In a regular pentagon, all sides are equal in length and each interior angle is 108 . A regular pentagon has five lines of reflectional symmetry , and rotational symmetry of order 5 through 72 , 144 , 216 and 288 . Its Schl fli symbol is 5 . The diagonal s of a regular pentagon are in golden ratio to its sides. The area of a regular convex pentagon with side length t is given by math A frac t 2 sqrt 25 10 sqrt 5 4 frac 5t 2 tan 54 circ 4 approx 1.720477401 t 2. math A pentagram or pentangle is a regular polygon regular star polygon star pentagon. Its Schl fli symbol is 5 2 . Its sides form the diagonals of a regular convex pentagon in this arrangement the Pentagram Golden ratio sides of the two pentagons are in the golden ratio . When a regular pentagon is Inscribed figure inscribed in a circle with radius R , its edge length t is given by the expression math t R sqrt frac 5 sqrt 5 2 2R sin 36 circ 2R sin frac pi 5 approx 1.17557050458 R. math Derivation of the area formula The area of any regular polygon is math A frac 1 2 Pa math where P is the perimeter of the polygon, a is the apothem . One can then substitute the respective values for P and a , which makes the formula math A frac 1 2 times frac 5t 1 times frac t tan 54 circ 2 math with t as the given side length. Then we can then rearrange the formula as math A frac 1 2 times frac 5t 2 tan 54 circ 2 math and then, we combine the two terms to get the final formula, which is math A frac 5t 2 tan 54 circ 4 . ma ... more details
About the number the album The Golden Ratio album calendar dates Golden number time File Golden ratio line.svg thumb Line segments in the golden ratio File SimilarGoldenRectangles.svg right thumb A golden rectangle with longer side span style color blue a span and shorter side span style color red b span , when placed adjacent to a square with sides of length span style color blue a span , will produce a Similarity geometry similar golden rectangle with longer side span style color green a b span and shorter side span style color blue a span . This illustrates the relationship math frac a b a frac a b equiv varphi math . In mathematics and the art s, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equality mathematics equal to the ratio of the larger quantity to the smaller one. The figure on the right illustrates the geometric relationship. Expressed algebraically math frac a b a frac a b equiv varphi, math where the Greek letter Phi letter phi math varphi math represents the golden ratio. Its value is math varphi frac 1 sqrt 5 2 1.61803 ,39887 ldots math . PLEASE DO NOT add additional decimal places here. There is long standing consensus that additional decimal places here do not contribute to understanding. Thank you. ref name quadform The golden ratio can be derived by the quadratic formula , by starting with the first number as 1, then solving for 2nd number x , where the ratios x 1 x x 1 or multiplying by x yields x 1 x sup 2 sup , or thus a quadratic equation x sup 2 sup x 1 0. Then, by the quadratic formula, for positive x b b sup 2 sup 4 ac 2 a with a 1, b 1, c 1, the solution for x is 1 1 sup 2 sup 4 1 1 2 1 or 1 5 2. ref At least since the 20th century , many artist s and architect s have proportioned their works to approximate the golden ratio e ... more details
Reg polyhedra db Reg polyhedron stat table D In geometry , a dodecahedron Greek , from , d deka twelve h dra base , seat or face is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant a Platonic solid . It is composed of 12 regular pentagon al faces, with three meeting at each vertex, and is represented by the Schl fli symbol 5,3 . It has 20 vertices and 30 edges. Its dual polyhedron is the icosahedron , with Schl fli symbol 3,5 . A large number of Other dodecahedra other nonregular polyhedra also have 12 sides, but are given other names. Other dodecahedra include the hexagonal bipyramid and the rhombic dodecahedron . Dimensions If the edge length of a regular dodecahedron is a , the radius of a circumscribed sphere one that touches the dodecahedron at all vertices is math r u a frac sqrt 3 4 left 1 sqrt 5 right approx 1.401258538 cdot a math OEIS2C A179296 and the radius of an inscribed sphere tangent to each of the dodecahedron s faces is math r i a frac 1 2 sqrt frac 5 2 frac 11 10 sqrt 5 approx 1.113516364 cdot a math while the midradius, which touches the middle of each edge, is math r m a frac 1 4 left 3 sqrt 5 right approx 1.309016994 cdot a math These quantities may also be expressed as math r u a , frac sqrt 3 2 varphi math math r i a , frac varphi 2 2 sqrt 3 varphi math math r m a , frac varphi 2 2 math where is the golden ratio . Note that, given a regular pentagonal dodecahedron of edge length one, r sub u sub is the radius of a circumscribing sphere about a cube of edge length , and r sub i sub is the apothem of a regular pentagon of edge length . Area and volume The surface area A and the volume V of a regular dodecahedron of edge length a are math A 3 sqrt 25 10 sqrt 5 a 2 approx 20.645728807a 2 math math V frac 1 4 15 7 sqrt 5 a 3 approx 7.6631189606a 3 math Orthogonal projections The dodecahedron has two special orthogonal projection s, centered, on vertices and pentagonal faces, correspond ... more details
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