This article needs additional citations for verification .(August 2014) (Learn how and when to remove this template message) |

In geometry an **equilateral pentagon** is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular (its five angles are equal; the measure is 108 degrees).

Four intersecting equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon. Each circle's center is one of four vertices of the pentagon. The remaining vertex is determined by one of the intersection points of the first and the last circle of the chain.

It is possible to describe the five angles of any convex equilateral pentagon with only two angles α and β, provided α ≥ β and δ is the smallest of the other angles. Thus the general equilateral pentagon can be regarded as a bivariate function *f(α, β)* where the rest of the angles can be obtained by using trigonometric relations. The equilateral pentagon described in this manner will be unique up to a rotation in the plane.

Regular pentagon (108° internal angles) | Regular star pentagram (36°) | Adjacent right angles (60°,150°,90°,90°,150°) |

Convex | Self-intersecting | Concave |

Degenerate into triangle (colinear edges) (~28.07°,0°,~75.96°,~75.96°,0°) | Degenerate (edge-vertex overlap) | Degenerate into trapezoid (colinear edges) (120°,120°,60°,0°,60°) |

When the equilateral pentagon is dissected into triangles, two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green). We assume that we are given the adjacent angles and .

According to the law of sines the length of the line dividing the green and blue triangles is:

The square of the length of the line dividing the orange and green triangles is:

According to the law of cosines, the cosine of δ can be seen from the figure:

Simplifying, δ is obtained as function of α and β:

The remaining angles of the pentagon can be found geometrically: The remaining angles of the orange and blue triangles are readily found by noting that two angles of an isosceles triangle are equal while all three angles sum to 180°. Then and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle is the sum of its three components, and the angle is the sum of its two components.

A cyclic pentagon is equiangular if and only if it has equal sides and thus is regular. Likewise, a tangential pentagon is equilateral if and only if it has equal angles and thus is regular.^{ [1] }

There are two infinite families of equilateral convex pentagons that tile the plane, one having two adjacent complementary angles and the other having two non-adjacent complementary angles. Some of these pentagons can tile in more than one way, and there is a sporadic example of an equilateral pentagon that can tile the plane but does not belong to either of these two families; its angles are 89°16', 144°32'30", 70°55', 135°22', and 99°54'30", none complementary.^{ [2] }

The equilateral pentagon as a function of two variables can be plotted in the two-dimensional plane. Each pair of values (α, β) maps to a single point of the plane and also maps to a single pentagon.

The periodicity of the values of α and β and the condition α ≥ β ≥ δ permit the size of the mapping to be limited. In the plane with coordinate axes α and β, **α = β** is a line dividing the plane in two parts (south border shown in orange in the drawing). **δ = β** as a curve divides the plane into different sections (north border shown in blue).

Both borders enclose a continuous region of the plane whose points map to unique equilateral pentagons. Points outside the region just map to repeated pentagons—that is, pentagons that when rotated or reflected can match others already described. Pentagons that map exactly onto those borders have a line of symmetry.

Inside the region of unique mappings there are three types of pentagons: stellated, concave and convex, separated by new borders.

The stellated pentagons have sides intersected by others. A common example of this type of pentagon is the pentagram. A condition for a pentagon to be stellated, or self-intersecting, is to have 2α + β ≤ 180°. So, in the mapping, the line **2α + β = 180°** (shown in orange at the north) is the border between the regions of stellated and non-stellated pentagons. Pentagons which map exactly to this border have a vertex touching another side.

The concave pentagons are non-stellated pentagons having at least one angle greater than 180°. The first angle which opens wider than 180° is γ, so **γ = 180°** (border shown in green at right) is a curve which is the border of the regions of concave pentagons and others, called convex. Pentagons which map exactly to this border have at least two consecutive sides appearing as a double length side, which resembles a pentagon degenerated to a quadrilateral.

The convex pentagons have all of their five angles smaller than 180° and no sides intersecting others. A common example of this type of pentagon is the regular pentagon.

In geometry, a **tetrahedron**, also known as a **triangular pyramid**, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

In plane Euclidean geometry, a **rhombus** is a quadrilateral whose four sides all have the same length. Another name is **equilateral quadrilateral**, since equilateral means that all of its sides are equal in length. The rhombus is often called a **diamond**, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a **lozenge**, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

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 trigonometry, the **law of tangents** is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.

In plane geometry, **Morley's trisector theorem** states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the **first Morley triangle** or simply the **Morley triangle**. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles.

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.

The **Wigner D-matrix** is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner. D stands for *Darstellung*, which means "representation" in German.

**Morrie's law** is a special trigonometric identity. Its name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.

In trigonometry, the **law of cosines** relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.

In trigonometry, **Mollweide's formula**, sometimes referred to in older texts as **Mollweide's equations**, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle.

**Solution of triangles** is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

**Hansen's problem** is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points *A* and *B*, and two unknown points *P*_{1} and *P*_{2}. From *P*_{1} and *P*_{2} an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of *P*_{1} and *P*_{2}. See figure; the angles measured are (*α*_{1}, *β*_{1}, *α*_{2}, *β*_{2}).

The **table of chords**, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's *Almagest*, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Centuries passed before more extensive trigonometric tables were created. One such table is the *Canon Sinuum* created at the end of the 16th century.

In trigonometry, the **law of cotangents** is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles.

**Stellar aberration** is an astronomical phenomenon "which produces an apparent motion of celestial objects". It can be proven mathematically that stellar aberration is due to the change of the astronomer's inertial frame of reference. The formula is derived with the use of Lorentz transformation of the star's *coordinates*.

**Pentagramma mirificum** is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book *Mirifici Logarithmorum Canonis Descriptio* along with rules that link the values of trigonometric functions of five parts of a right spherical triangle. The properties of *pentagramma mirificum* were studied, among others, by Carl Friedrich Gauss.

The **equal detour point** is a triangle center with the Kimberling number X(176). It is characterized by the equal detour property, that is if you travel from any vertex of a triangle to another by taking a detour through some inner point then the additional distance travelled is constant. This means the following equation has to hold:

- ↑ De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.
- ↑ Schattschneider, Doris (1978), "Tiling the plane with congruent pentagons",
*Mathematics Magazine*,**51**(1): 29–44, doi:10.1080/0025570X.1978.11976672, JSTOR 2689644, MR 0493766

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

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

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

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