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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 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 ...
or 5-gon. The sum of the internal angles 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.


Regular pentagons

A '' regular pentagon'' has Schläfli symbol and interior angles 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 th ...
, 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 ...
of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio 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 incen ...
(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. ...
). 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 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 incen ...
. 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. ...
, 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. 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. 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''. 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 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: :\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. 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 value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
. 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 cons ...
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, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
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 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 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 r ...
. 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 (mathematics), 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 ...
there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z5, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway 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 edges.


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.


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 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 so ...
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 o ...
whose coefficients are functions of the sides of the pentagon. There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons. 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”''


Pentagons in tiling

A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a regular tiling (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 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 a ...
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. File:Morning Glory Flower.jpg, Morning glories, 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) '' pist ...
of an
apple An apple is an edible fruit produced by an apple tree (''Malus domestica''). Apple trees are cultivated worldwide and are the most widely grown species in the genus '' Malus''. The tree originated in Central Asia, where its wild ancest ...
contains five carpels, arranged in a five-pointed star File:Carambola Starfruit.jpg, Starfruit is another fruit with fivefold symmetry.


Animals

File:Oreaster reticulatus201905mx.jpg, A sea star. Many
echinoderms An echinoderm () is any member of the phylum Echinodermata (). The adults are recognisable by their (usually five-point) radial symmetry, and include starfish, brittle stars, sea urchins, sand dollars, and sea cucumbers, as well as the ...
have fivefold radial symmetry. File:Sea Urchin Endoskeleton.jpg, Another example of echinoderm, a sea urchin endoskeleton. File:Haeckel Ophiodea.jpg, An illustration of brittle stars, also echinoderms with a pentagonal shape.


Minerals

File:Ho-Mg-ZnQuasicrystal.jpg, A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons. File:Pyrite elbe.jpg, A pyritohedral 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 sec ...
. File:Home base of baseball field in Třebíč, Třebíč District.jpg, Home plate of a baseball field


See also

* Associahedron; A pentagon is an order-4 associahedron * Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal faces * Golden ratio *
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 numbers * Pentagram * Pentagram map * Pentastar, 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