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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a pentagon (from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
or 5-gon. The sum of the
internal angle 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 ) if ...
s in a
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
pentagon'') is called a
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
.


Regular pentagons

A '' regular pentagon'' has Schläfli symbol and
interior angle 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 ) if ...
s of 108°. A '' regular pentagon'' has five lines of
reflectional symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...
, and
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
of order 5 (through 72°, 144°, 216° and 288°). The
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s of a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
regular pentagon are in the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqrt \cdot H\approx 1.051~H, \\ R &= \sqrt t\approx 0.8507~t, \\ D &= R\ = 2R\cos 18^\circ = 2R\cos\frac \approx 1.902~R. \end The area of a convex regular pentagon with side length t is given by :\begin A &= \frac = \frac \\ &= \frac \approx 1.720~t^2. \end If the circumradius R of a regular pentagon is given, its edge length t is found by the expression :t = R\ = 2R\sin 36^\circ = 2R\sin\frac \approx 1.176~R, and its area is :A = \frac\sqrt; since the area of the circumscribed circle is \pi R^2, the regular pentagon fills approximately 0.7568 of its circumscribed circle.


Derivation of the area formula

The area of any regular polygon is: :A = \fracPr where ''P'' is the perimeter of the polygon, and ''r'' is the
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be 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. ...
(equivalently the
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. T ...
). Substituting the regular pentagon's values for ''P'' and ''r'' gives the formula :A = \frac \cdot 5t \cdot \frac = \frac with side length ''t''.


Inradius

Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The
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. T ...
, which is the radius ''r'' of the inscribed circle, of a regular pentagon is related to the side length ''t'' by :r = \frac = \frac \approx 0.6882 \cdot t.


Chords from the circumscribed circle to the vertices

Like every regular convex polygon, the regular convex pentagon has a
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.


Point in plane

For an arbitrary point in the plane of a regular pentagon with circumradius R, whose distances to the centroid of the regular pentagon and its five vertices are L and d_i respectively, we have :\begin \textstyle \sum_^5 d_i^2 &= 5\left(R^2 + L^2\right), \\ \textstyle \sum_^5 d_i^4 &= 5\left(\left(R^2 + L^2\right)^2 + 2R^2 L^2\right), \\ \textstyle \sum_^5 d_i^6 &= 5\left(\left(R^2 + L^2\right)^3 + 6R^2 L^2 \left(R^2 + L^2\right)\right), \\ \textstyle \sum_^5 d_i^8 &= 5\left(\left(R^2 + L^2\right)^4 + 12R^2 L^2 \left(R^2 + L^2\right)^2 + 6R^4 L^4\right). \end If d_i are the distances from the vertices of a regular pentagon to any point on its circumcircle, then :3\left(\textstyle \sum_^5 d_i^2\right)^2 = 10 \textstyle \sum_^5 d_i^4 .


Geometrical constructions

The regular pentagon is constructible with compass and straightedge, as 5 is a
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
. A variety of methods are known for constructing a regular pentagon. Some are discussed below.


Richmond's method

One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's ''
Polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
''. The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point ''C'' and a midpoint ''M'' is marked halfway along its radius. This point is joined to the periphery vertically above the center at point ''D''. Angle ''CMD'' is bisected, and the bisector intersects the vertical axis at point ''Q''. A horizontal line through ''Q'' intersects the circle at point ''P'', and chord ''PD'' is the required side of the inscribed pentagon. To determine the length of this side, the two right triangles ''DCM'' and ''QCM'' are depicted below the circle. Using
Pythagoras' theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
and two sides, the hypotenuse of the larger triangle is found as \scriptstyle \sqrt/2. Side ''h'' of the smaller triangle then is found using the
half-angle formula In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
: :\tan(\phi/2) = \frac \ , where cosine and sine of ''ϕ'' are known from the larger triangle. The result is: :h = \frac \ . If DP is truly the side of a regular pentagon, m \angle\mathrm = 54^\circ, so DP = 2 cos(54°), QD = DP cos(54°) = 2cos2(54°), and CQ = 1 − 2cos2(54°), which equals −cos(108°) by the cosine
double angle formula In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
. This is the cosine of 72°, which equals \left(\sqrt 5 - 1\right)/4 as desired.


Carlyle circles

The Carlyle circle was invented as a geometric method to find the roots of a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
. This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows: # Draw a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
in which to inscribe the pentagon and mark the center point ''O''. # Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point ''B''. # Construct a vertical line through the center. Mark one intersection with the circle as point ''A''. # Construct the point ''M'' as the midpoint of ''O'' and ''B''. # Draw a circle centered at ''M'' through the point ''A''. Mark its intersection with the horizontal line (inside the original circle) as the point ''W'' and its intersection outside the circle as the point ''V''. # Draw a circle of radius ''OA'' and center ''W''. It intersects the original circle at two of the vertices of the pentagon. # Draw a circle of radius ''OA'' and center ''V''. It intersects the original circle at two of the vertices of the pentagon. # The fifth vertex is the rightmost intersection of the horizontal line with the original circle. Steps 6–8 are equivalent to the following version, shown in the animation: : 6a. Construct point F as the midpoint of O and W. : 7a. Construct a vertical line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex is the rightmost intersection of the horizontal line with the original circle. : 8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.


Euclid's method

A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
in his '' Elements'' circa 300 BC.


Physical construction methods

* A regular pentagon may be created from just a strip of paper by tying an
overhand knot The overhand knot is one of the most fundamental knots, and it forms the basis of many others, including the simple noose, overhand loop, angler's loop, reef knot, fisherman's knot, Half hitch, and water knot. The overhand knot is a stopper, ...
into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
when backlit. * Construct a regular
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
on stiff paper or card. Crease along the three diameters between opposite vertices. Cut from one vertex to the center to make an equilateral triangular flap. Fix this flap underneath its neighbor to make a
pentagonal pyramid In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regu ...
. The base of the pyramid is a regular pentagon.


Symmetry

The ''regular pentagon'' has Dih5 symmetry, order 10. Since 5 is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
there is one subgroup with dihedral symmetry: Dih1, and 2
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
symmetries: Z5, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon.
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
labels these by a letter and group order. Full symmetry of the regular form is r10 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as
directed edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
s.


Regular pentagram

A pentagram or pentangle is a regular
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
pentagon. Its Schläfli symbol is . Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
.


Equilateral pentagons

An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
similarity, because it is equilateral and it is equiangular (its five angles are equal).


Cyclic pentagons

A
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth the square root of one of the roots of a
septic equation In algebra, a septic equation is an equation of the form :ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\, where . A septic function is a function of the form :f(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\, where . In other words, it is a polynomial of ...
whose coefficients are functions of the sides of the pentagon. There exist cyclic pentagons with rational sides and rational area; these are called
Robbins pentagon In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers. History Robbins pentagons were named by after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a ...
s. It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational.


General convex pentagons

For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.''Inequalities proposed in “
Crux Mathematicorum ''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk. The journ ...
”''


Pentagons in tiling

A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a
regular tiling Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...
(one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons: The maximum known
packing density A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. I ...
of a regular pentagon is approximately 0.921, achieved by the
double lattice In mathematics, especially in geometry, a double lattice in is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of an ...
packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that the double lattice packing of the regular pentagon (which they call the "pentagonal ice-ray" packing, and which they trace to the work of Chinese artisans in 1900) has the optimal density among all packings of regular pentagons in the plane. , their proof has not yet been refereed and published. There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around the pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all . To find the number of sides this polygon has, the result is , which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons. There are 15 classes of pentagons that can monohedrally tile the plane. None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry.


Pentagons in polyhedra


Pentagons in nature


Plants

File:BhindiCutUp.jpg, Pentagonal cross-section of
okra Okra or Okro (, ), ''Abelmoschus esculentus'', known in many English-speaking countries as ladies' fingers or ochro, is a flowering plant in the mallow family. It has edible green seed pods. The geographical origin of okra is disputed, with su ...
. File:Morning Glory Flower.jpg,
Morning glories Morning glory (also written as morning-glory) is the common name for over 1,000 species of flowering plants in the family Convolvulaceae, whose current taxonomy and systematics are in flux. Morning glory species belong to many genera, some of ...
, like many other flowers, have a pentagonal shape. File:Sterappel dwarsdrsn.jpg, The
gynoecium Gynoecium (; ) is most commonly used as a collective term for the parts of a flower that produce ovules and ultimately develop into the fruit and seeds. The gynoecium is the innermost whorl of a flower; it consists of (one or more) ''pistils'' ...
of an
apple An apple is an edible fruit produced by an apple tree (''Malus domestica''). Apple fruit tree, trees are agriculture, cultivated worldwide and are the most widely grown species in the genus ''Malus''. The tree originated in Central Asia, wh ...
contains five carpels, arranged in a
five-pointed star A five-pointed star (☆), geometrically an equilateral concave decagon, is a common ideogram in modern culture. Comparatively rare in classical heraldry, it was notably introduced for the flag of the United States in the Flag Act of 1777 and s ...
File:Carambola Starfruit.jpg,
Starfruit Carambola, also known as star fruit, is the fruit of '' Averrhoa carambola'', a species of tree native to tropical Southeast Asia. The mildly poisonous fruit is commonly consumed in parts of Brazil, Southeast Asia, South Asia, the South Pacif ...
is another fruit with fivefold symmetry.


Animals

File:Oreaster reticulatus201905mx.jpg, A
sea star Starfish or sea stars are star-shaped echinoderms belonging to the class Asteroidea (). Common usage frequently finds these names being also applied to ophiuroids, which are correctly referred to as brittle stars or basket stars. Starfish a ...
. Many echinoderms have fivefold radial symmetry. File:Sea Urchin Endoskeleton.jpg, Another example of echinoderm, a
sea urchin Sea urchins () are spiny, globular echinoderms in the class Echinoidea. About 950 species of sea urchin live on the seabed of every ocean and inhabit every depth zone from the intertidal seashore down to . The spherical, hard shells (tests) of ...
endoskeleton. File:Haeckel Ophiodea.jpg, An illustration of
brittle stars Brittle stars, serpent stars, or ophiuroids (; ; referring to the serpent-like arms of the brittle star) are echinoderms in the class Ophiuroidea, closely related to starfish. They crawl across the sea floor using their flexible arms for locomo ...
, also echinoderms with a pentagonal shape.


Minerals

File:Ho-Mg-ZnQuasicrystal.jpg, A Ho-Mg-Zn icosahedral
quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...
formed as a pentagonal
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
. The faces are true regular pentagons. File:Pyrite elbe.jpg, A
pyritohedral image:tetrahedron.jpg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that c ...
crystal of
pyrite The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Iron, FeSulfur, S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral. Pyrite's metallic Luster (mineralogy), lust ...
. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.


Other examples

File:The Pentagon January 2008.jpg,
The Pentagon The Pentagon is the headquarters building of the United States Department of Defense. It was constructed on an accelerated schedule during World War II. As a symbol of the U.S. military, the phrase ''The Pentagon'' is often used as a metony ...
, headquarters of the
United States Department of Defense The United States Department of Defense (DoD, USDOD or DOD) is an executive branch department of the federal government charged with coordinating and supervising all agencies and functions of the government directly related to national secu ...
. File:Home base of baseball field in Třebíč, Třebíč District.jpg,
Home plate A baseball field, also called a ball field or baseball diamond, is the field upon which the game of baseball is played. The term can also be used as a metonym for a baseball park. The term sandlot is sometimes used, although this usually refers ...
of a
baseball field A baseball field, also called a ball field or baseball diamond, is the field upon which the game of baseball is played. The term can also be used as a metonym for a baseball park. The term sandlot is sometimes used, although this usually refers ...


See also

*
Associahedron In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of ...
; A pentagon is an order-4 associahedron *
Dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
, a polyhedron whose regular form is composed of 12 pentagonal faces *
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
*
List of geometric shapes Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools. Mathematics * List of mathematical shapes * List of two- ...
*
Pentagonal number A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
s *
Pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
*
Pentagram map In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a ne ...
*
Pentastar The Chrysler Pentastar engine family is a series of aluminium ( die-cast cylinder block) dual overhead cam 24-valve gasoline V6 engines introduced for model-year 2011 Chrysler, Dodge and Jeep vehicles. The engine was initially named " Phoenix ...
, the Chrysler logo * Pythagoras' theorem#Similar figures on the three sides * Trigonometric constants for a pentagon


In-line notes and references


External links

*
Animated demonstration
constructing an inscribed pentagon with compass and straightedge.
How to construct a regular pentagon
with only a compass and straightedge.

using only a strip of paper

with interactive animation
Renaissance artists' approximate constructions of regular pentagons


How to calculate various dimensions of regular pentagons. {{Polytopes Constructible polygons Polygons by the number of sides 5 (number) Elementary shapes