Pentagon | |
---|---|

An equilateral pentagon, i.e. a pentagon whose five sides all have the same length | |

Edges and vertices | 5 |

Internal angle (degrees) | 108° (if equiangular, including regular) |

In geometry, a **pentagon** (from the Greek πέντε *pente* and γωνία *gonia*, meaning *five* and *angle*^{ [1] }) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

- Regular pentagons
- Derivation of the area formula
- Inradius
- Chords from the circumscribed circle to the vertices
- Point in plane
- Construction of a regular pentagon
- Euclid's method
- Physical methods
- Symmetry
- Equilateral pentagons
- Cyclic pentagons
- General convex pentagons
- Graphs
- Examples of pentagons
- Plants
- Animals
- Minerals
- Artificial
- Pentagons in tiling
- Pentagons in polyhedra
- See also
- In-line notes and references
- External links

A pentagon may be simple or self-intersecting. A self-intersecting *regular pentagon* (or * star pentagon*) is called a pentagram.

Regular pentagon | |
---|---|

A regular pentagon | |

Type | Regular polygon |

Edges and vertices | 5 |

Schläfli symbol | {5} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{5}), order 2×5 |

Internal angle (degrees) | 108° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

A * regular pentagon* has Schläfli symbol {5} and interior angles are 108°.

A * regular pentagon* has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Its height (distance from one side to the opposite vertex) and width (distance between two farthest separated points, which equals the diagonal length) are given by

where *R* is the radius of the circumcircle.

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

A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.

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

and its area is

since the area of the circumscribed circle is the regular pentagon fills approximately 0.7568 of its circumscribed circle.

The area of any regular polygon is:

where *P* is the perimeter of the polygon, and *r* is the inradius (equivalently the apothem). Substituting the regular pentagon's values for *P* and *r* gives the formula

with side length *t*.

Like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius *r* of the inscribed circle, of a regular pentagon is related to the side length *t* by

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.

For an arbitrary point in the plane of a regular pentagon with circumradius , whose distances to the centroid of the regular pentagon and its five vertices are and respectively, we have ^{ [2] }

If are the distances from the vertices of a regular pentagon to any point on its circumcircle, then ^{ [2] }

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.

One method to construct a regular pentagon in a given circle is described by Richmond^{ [3] } and further discussed in Cromwell's * Polyhedra *.^{ [4] }

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 . Side *h* of the smaller triangle then is found using the half-angle formula:

where cosine and sine of *ϕ* are known from the larger triangle. The result is:

With this side known, attention turns to the lower diagram to find the side *s* of the regular pentagon. First, side *a* of the right-hand triangle is found using Pythagoras' theorem again:

Then *s* is found using Pythagoras' theorem and the left-hand triangle as:

The side *s* is therefore:

a well-established result.^{ [5] } Consequently, this construction of the pentagon is valid.

The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation.^{ [6] } This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:^{ [7] }

- Draw a circle 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.

- We first note that a regular pentagon can be divided into 10 congruent triangles as shown in the
**Observation**. Also, cos 36° = .**†** - In
**Step 1**, we use four units (shown in blue) and a right angle to construct a segment of length 1+√5, specifically by creating a 1-2-√5 right triangle and then extending the hypotenuse of √5 by a length of 1. We then bisect that segment – and then bisect again – to create a segment of length (shown in red.) - In
**Step 2**, we construct two concentric circles centered at*O*with radii of length 1 and length . We then place*P*arbitrarily on the smaller circle, as shown. Constructing a line perpendicular to OP passing through*P*, we construct the first side of the pentagon by using the points created at the intersection of the tangent and the unit circle. Copying that length four times along the outer edge of the unit circles gives us our regular pentagon.

- (using the angle addition formula for cosine)
- (using double and half angle formulas)

- Let
*u*= cos 36°. First, note that 0 <*u*< 1 (which will help us simplify as we work). Now,

This follows quickly from the knowledge that twice the sine of 18 degrees is the reciprocal golden ratio, which we know geometrically from the triangle with angles of 72,72,36 degrees. From trigonometry, we know that the cosine of twice 18 degrees is 1 minus twice the square of the sine of 18 degrees, and this reduces to the desired result with simple quadratic arithmetic.

The regular pentagon according to the golden ratio, dividing a line segment by exterior division

- Draw a segment
*AB*whose length is the given side of the pentagon. - Extend the segment
*BA*from point*A*about three quarters of the segment*BA*. - Draw an arc of a circle, centre point
*B*, with the radius*AB*. - Draw an arc of a circle, centre point
*A*, with the radius*AB*; there arises the intersection*F*. - Construct a perpendicular to the segment
*AB*through the point*F*; there arises the intersection*G*. - Draw a line parallel to the segment
*FG*from the point*A*to the circular arc about point*A*; there arises the intersection*H*. - Draw an arc of a circle, centre point
*G*with the radius*GH*to the extension of the segment*AB*; there arises the intersection*J*. - Draw an arc of a circle, centre point
*B*with the radius*BJ*to the perpendicular at point*G*; there arises the intersection*D*on the perpendicular, and the intersection*E*with the circular arc that was created about the point*A*. - Draw an arc of a circle, centre point
*D*, with the radius*BA*until this circular arc cuts the other circular arc about point*B*; there arises the intersection*C*. - Connect the points
*BCDEA*. This results in the pentagon.

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 in his * Elements * circa 300 BC.^{ [8] }^{ [9] }

A direct method using degrees follows:

- Draw a circle and choose a point to be the pentagon's (e.g. top center)
- Choose a point
*A*on the circle that will serve as one vertex of the pentagon. Draw a line through*O*and*A*. - Draw a guideline through it and the circle's center
- Draw lines at 54° (from the guideline) intersecting the pentagon's point
- Where those intersect the circle, draw lines at 18° (from parallels to the guideline)
- Join where they intersect the circle

After forming a regular convex pentagon, if one joins the non-adjacent corners (drawing the diagonals of the pentagon), one obtains a pentagram, with a smaller regular pentagon in the center. Or if one extends the sides until the non-adjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on the accuracy of the protractor used to measure the angles.

- 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 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. The base of the pyramid is a regular pentagon.

The *regular pentagon* has Dih_{5} symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih_{1}, and 2 cyclic group symmetries: Z_{5}, and Z_{1}.

These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order.^{ [10] } 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.

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).

A cyclic 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 whose coefficients are functions of the sides of the pentagon.^{ [11] }^{ [12] }^{ [13] }

There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons. In a Robbins pentagon, either all diagonals are rational or all are irrational, and it is conjectured that all the diagonals must be rational.^{ [14] }

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.^{ [15] }^{:p.75,#1854}

The K_{5} complete graph is often drawn as a *regular pentagon* with all 10 edges connected. This graph also represents an orthographic projection of the 5 vertices and 10 edges of the 5-cell. The rectified 5-cell, with vertices at the mid-edges of the 5-cell is projected inside a pentagon.

5-cell (4D) | Rectified 5-cell (4D) |

- Pentagonal cross-section of okra.
- Morning glories, like many other flowers, have a pentagonal shape.
- Starfruit is another fruit with fivefold symmetry.

- A sea star. Many echinoderms have fivefold radial symmetry.
- Another example of echinoderm, a sea urchin endoskeleton.
- An illustration of brittle stars, also echinoderms with a pentagonal shape.

- A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons.
- A pyritohedral crystal of pyrite. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.

- The Pentagon, headquarters of the United States Department of Defense.
- Home plate of a baseball field

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 360° / 108° = 3^{1}⁄_{3} (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 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.^{ [16] }As of 2020^{ [update] }, 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 (360 − 108) / 2 = 126°. To find the number of sides this polygon has, the result is 360 / (180 − 126) = 6^{2}⁄_{3}, 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.

1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

I_{h} | T_{h} | T_{d} | O | I | D_{5d} |
---|---|---|---|---|---|

Dodecahedron | Pyritohedron | Tetartoid | Pentagonal icositetrahedron | Pentagonal hexecontahedron | Truncated trapezohedron |

- 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
- Pentagonal numbers
- Pentagram
- Pentagram map
- Pentastar, the Chrysler logo
- Pythagoras' theorem#Similar figures on the three sides
- Trigonometric constants for a pentagon

- ↑ "pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 August 2014.
- 1 2 Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids".
*Communications in Mathematics and Applications*.**11**: 335–355. - ↑ Herbert W Richmond (1893). "Pentagon".
- ↑ Peter R. Cromwell.
*Polyhedra*. p. 63. ISBN 0-521-66405-5. - ↑ This result agrees with Herbert Edwin Hawkes; William Arthur Luby; Frank Charles Touton (1920). "Exercise 175".
*Plane geometry*. Ginn & Co. p. 302. - ↑ Eric W. Weisstein (2003).
*CRC concise encyclopedia of mathematics*(2nd ed.). CRC Press. p. 329. ISBN 1-58488-347-2. - ↑ DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF).
*The American Mathematical Monthly*.**98**(2): 97–108. doi:10.2307/2323939. Archived from the original (PDF) on 2015-12-21. - ↑ George Edward Martin (1998).
*Geometric constructions*. Springer. p. 6. ISBN 0-387-98276-0. - ↑
*Euklid's Elements of Geometry, Book 4, Proposition 11*(PDF). Translated by Richard Fitzpatrick. 2008. p. 119. ISBN 978-0-6151-7984-1. - ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
- ↑ Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource.
- ↑ Robbins, D. P. (1994). "Areas of Polygons Inscribed in a Circle".
*Discrete and Computational Geometry*.**12**: 223–236. doi: 10.1007/bf02574377 . - ↑ Robbins, D. P. (1995). "Areas of Polygons Inscribed in a Circle".
*The American Mathematical Monthly*.**102**: 523–530. doi:10.2307/2974766. - ↑
- Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area",
*Journal of Number Theory*,**128**(1): 17–48, doi:10.1016/j.jnt.2007.05.005, MR 2382768, archived from the original on 2018-11-12, retrieved 2012-03-29.

- Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area",
- ↑
*Inequalities proposed in “Crux Mathematicorum”*, . - ↑ Hales, Thomas; Kusner, Wöden (September 2016),
*Packings of regular pentagons in the plane*, arXiv: 1602.07220

Look up in Wiktionary, the free dictionary. pentagon |

- Weisstein, Eric W. "Pentagon".
*MathWorld*. - Animated demonstration constructing an inscribed pentagon with compass and straightedge.
- How to construct a regular pentagon with only a compass and straightedge.
- How to fold a regular pentagon using only a strip of paper
- Definition and properties of the pentagon, with interactive animation
- Renaissance artists' approximate constructions of regular pentagons
- Pentagon. How to calculate various dimensions of regular pentagons.

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

A **quadrilateral** is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include **quadrangle**, **tetragon**, and **4-gon**. A quadrilateral with vertices , , and is sometimes denoted as .

In geometry, a **hexagon** is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In geometry, the **incircle** or **inscribed circle** of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

In geometry, an **equilateral triangle** is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a **regular triangle**.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or circumscribed circle, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In geometry, a **heptadecagon** or 17-gon is a seventeen-sided polygon.

In geometry, a **decagon** is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In Euclidean geometry, a **regular polygon** is a polygon that is equiangular and equilateral. Regular polygons may be either **convex** or **star**. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In geometry, a **heptagon** is a seven-sided polygon or 7-gon.

In geometry, an **icosagon** or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

In geometry an **enneadecagon** or **enneakaidecagon** or 19-gon is a nineteen-sided polygon.

In geometry, a **tridecagon** or **triskaidecagon** or 13-gon is a thirteen-sided polygon.

In geometry, a **square** is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices *ABCD* would be denoted *ABCD*.

In geometry, a **pentadecagon** or **pentakaidecagon** or 15-gon is a fifteen-sided polygon.

In Euclidean geometry, **Ptolemy's theorem** is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

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**.

In geometry, a **tetradecagon** or **tetrakaidecagon** or 14-gon is a fourteen-sided polygon.

In geometry, an **octacontagon** is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees.

In geometry, a **tetracontagon** or **tessaracontagon** is a forty-sided polygon or 40-gon. The sum of any tetracontagon's interior angles is 6840 degrees.

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Images, videos and audio are available under their respective licenses.