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 .... br Properties of apothems The apothem a can be used to find the area of any regular ... is equal to the apothem multiplied by half the perimeter since ns p . math A frac nsa 2 frac pa 2 ... 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 ... circ 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 ..., Apothem, and Chord by Ed Pegg, Jr. , The Wolfram Demonstrations Project . Category Geometry Category ... more details
Summary Regular square pyramid showing the median triangle with apothem a , semi base b , height h , and inclination angle. Licensing PD self date October 2006 ... more details
it s just 2 times apothem for even n, or apothem radius for odd n. Is it worth recording? Radius ..., s or apothem , a math r frac s 2 sin frac pi n frac a cos frac pi n math Area This section is linked ... with side geometry side s , apothem a , and circumradius r The area A of a convex regular n sided polygon having side geometry side s , apothem a , perimeter p , and circumradius r is given by ref ... 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 or axis of symmetry 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. The plural in English language English is radii as in Latin , but radiuses can be used, though it rarely is. ref name radic http dictionary.reference.com browse Radius Definition ... more details
mathematical constant The area of a disk mathematics disk the region inside 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 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. times 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. 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 Hill, George. http books.google.com books?id zSMAAAAAYAAJ&lpg PA124&ots Z973CjYQgF&dq 22the 20greater 20the 20number 20of 20sides 20of 20the 20polygon 22 20and 20 22George 20Anthony 20Hill 22&pg PA124 v onepage&q&f false Lessons in Geometry For the Use of Beginners , page 124 1894 . ref Archimedes s proof Following Harvtxt Archimedes c. 260 BCE , compare a circle to a right triangle whose base has the length of the circle s circumference and whose height equals the circle s radius. If the area of the circle is not equal to that of the tr ... 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 A regular pentagon has all sides of equal length and all interior angles are equal measure 108 . It 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 Chord geometry chords 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 &ndash 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 math t math 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 ... more details
, whose edges are the pyramid s apothem a , semi base b , and height h the face inclination angle ... the apothem slant height along the bisector of a face is equal to times the semi base half ... more details
Reg polyhedra db Reg polyhedron stat table D In geometry , a dodecahedron Greek , from twelve 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. The most frequently named other dodecahedron is 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 frac a 4 left sqrt 15 sqrt 3 right approx 1.401258538 cdot a math and the radius of an inscribed sphere tangent to each of the dodecahedron s faces is math r i frac a 20 sqrt 250 110 sqrt 5 approx 1.113516364 cdot a math while the midradius, which touches the middle of each edge, is math r m frac a 4 left 3 sqrt 5 right approx 1.309016994 cdot a math These quantities may also be expressed as math r u frac sqrt 3 2 phi , a math math r i frac phi 2 2 sqrt 3 phi , a , math math r m frac phi 2 2 a math where math phi , math is the golden ratio . Note that, given a regular pentagonal dodecahedron of edge length one, math r u , math is the radius of a circumscribing sphere about a cube of edge length math phi , math , and math r i , math is the apothem of a regular pentagon of edge length math phi , math . 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 Cartesian coordinates The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin 1, 1, 1 0, 1 , 1 , , 0 ... more details
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