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Pentagon
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a pentagon is any five-sided polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
. A pentagon may be simple or self-intersecting. The internal angle
Internal angle

In geometry, an interior angle is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon....
s in a simple pentagon total 540°.

term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to 108°). Its Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is .

The area of a regular convex pentagon with side length t is given by



A pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
 or pentangle is a regular
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 star
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
 pentagon.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a pentagon is any five-sided polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
. A pentagon may be simple or self-intersecting. The internal angle
Internal angle

In geometry, an interior angle is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon....
s in a simple pentagon total 540°.

Regular pentagons

The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to 108°). Its Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is .

The area of a regular convex pentagon with side length t is given by



A pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
 or pentangle is a regular
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 star
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
 pentagon. Its Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is . Its sides form the diagonals of a regular convex pentagon - in this arrangement the sides of the two pentagons
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
 are in the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
.

When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression

.


Construction


A regular pentagon is constructible
Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be Compass and straightedge constructions. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....
 using a compass and straightedge
Compass and straightedge

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an Idealization ruler and Compass ....
, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid
Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
 in his Elements
Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
 circa 300 BC.

One method to construct a regular pentagon in a given circle is as follows:
Pentagon Construct
An alternative method is this:

Pentagon Construction
  1. Draw a circle
    Circle

    A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
     in which to inscribe the pentagon and mark the center point O. (This is the green circle in the diagram to the right).
  2. Choose a point A on the circle that will serve as one vertex of the pentagon. Draw a line through O and A.
  3. Construct a line perpendicular to the line OA passing through O. Mark its intersection with one side of the circle as the point B.
  4. Construct the point C as the midpoint of O and B.
  5. Draw a circle centered at C through the point A. Mark its intersection with the line OB (inside the original circle) as the point D.
  6. Draw a circle centered at A through the point D. Mark its intersections with the original (green) circle as the points E and F.
  7. Draw a circle centered at E through the point A. Mark its other intersection with the original circle as the point G.
  8. Draw a circle centered at F through the point A. Mark its other intersection with the original circle as the point H.
  9. Construct the regular pentagon AEGHF.


After forming a regular convex pentagon, if you join the non-adjacent corners (drawing the diagonals of the pentagon), you obtain a pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
, with a smaller regular pentagon in the center. Or if you extend the sides until the non-adjacent ones meet, you obtain a larger pentagram.

A simple method of creating a regular pentagon from just a strip of paper is by tying an overhand knot
Overhand knot

The overhand knot is one of the most fundamental knots and forms the basis of many others including the Noose, overhand loop, angler's loop, fisherman's knot and water 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
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
 when backlit.

Pentagons in nature


Plants



Animals


See also

  • The Pentagon
    The Pentagon

    The Pentagon is the headquarters of the United States Department of Defense, located in Arlington County, Virginia, Virginia. As a symbol of the Military of the United States, "the Pentagon" is often used Metonymy to refer to the Department of Defense rather than the building itself....
    , headquarters for the U.S. Department of Defense
  • Dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
    , a polyhedron whose regular form is composed of 12 pentagonal faces
  • Trigonometric constants for a pentagon
    Exact trigonometric constants

    Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into Radical forms which allow further simplification....
  • Pentagonal number
    Pentagonal number

    A pentagonal number is a figurate number that extends the concept of triangular number and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotational symmetry....
    s
  • Associahedron
    Associahedron

    In mathematics, an associahedron or Stasheff polytope Kn is a convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of n letters and the edges correspond to single application of the associativity rule....
     - A pentagon is an order-4 associahedron
  • Pentagram
    Pentagram

    A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
  • Pentastar, the Chrysler logo


External links

  • with only a only compass and straightedge.
  • using only a strip of paper
  • , with interactive animation
  • at