Pentagon

Pentagon
J Hokkanen · CC BY-SA 3.0 · source
Name	Pentagon
Caption	A regular pentagon
Schläfli	{5}
Angle	108° (for a regular pentagon)
Dual	Self-dual

Pentagon. In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting, with the most studied and symmetric form being the regular pentagon, a fundamental shape in geometry with deep connections to the golden ratio. Its properties have fascinated mathematicians from the time of the ancient Greeks and it appears in various contexts, from the design of the United States Department of Defense headquarters to patterns in the natural world.

Geometry and properties

A pentagon is defined by five vertices and five edges. For any simple pentagon, the sum of its interior angles is always 540 degrees, a property derived from the general formula for polygon angle sums. Important elements of a pentagon include its diagonals; a convex pentagon has five diagonals, and these diagonals intersect in ways that create smaller internal polygons. The study of cyclic pentagons, those that can be inscribed in a circle, involves complex trigonometric relationships explored by mathematicians like Albrecht Dürer. The area of an irregular pentagon can be calculated through triangulation, dividing it into three triangles, or by applying the shoelace formula using the coordinates of its vertices.

Regular pentagon

A regular pentagon is both equilateral and equiangular, with all sides equal and all internal angles measuring 108°. Its side length, circumradius, inradius, and area are all interrelated through the golden ratio, φ. The length of a diagonal in a regular pentagon is φ times the length of a side. This relationship means the pentagon is rich in golden ratios, and the pentagram formed by its diagonals is a classic symbol of this proportion. The area of a regular pentagon can be expressed using its side length and trigonometric functions of 36° and 72°, angles central to the geometry of the regular decagon. The regular pentagon's construction was a significant problem in antiquity, closely tied to the discovery of irrational numbers.

Construction

The classical construction of a regular pentagon using only a compass and straightedge was solved by ancient Greek geometers and is detailed in Euclid's Elements. This method relies on constructing an isosceles triangle with base angles twice the vertex angle, which inherently involves the golden ratio. Alternative methods include Richmond's method, which provides a more efficient modern construction, and approximations used in technical drawing. The problem is also connected to the construction of the regular decagon, as inscribing a pentagon in a given circle is equivalent to constructing a decagon. The impossibility of trisecting the angle with compass and straightedge alone is related to the cubic equations arising from pentagon construction.

Symmetry

The regular pentagon has dihedral symmetry of order 10, denoted Dih₅. This includes five rotational symmetries and five reflection symmetries, with its symmetry group being the dihedral group of order 10. These symmetries permute the five vertices and are fundamental in group theory applications. The five reflection axes each pass through a vertex and the midpoint of the opposite side. The rotational symmetry is evident in objects like the United States Great Seal and the Baha'i Houses of Worship. The pentagon's symmetry is also a two-dimensional example of the symmetry found in the Platonic solid known as the dodecahedron.

Pentagons in nature and culture

Pentagonal symmetry appears in various natural forms, most notably in the starfish and the arrangement of petals in many flowers like the morning glory. The organic compound ferrocene has a pentagonal molecular geometry, and the Vitruvian Man by Leonardo da Vinci is often inscribed in a pentagon. In architecture, the shape is famously used for the headquarters of the United States Department of Defense in Arlington. The pentagon features prominently in symbolism, from the pentagram in Western esotericism to its use on the flag of Morocco and in the design of the Bent Pyramid in Ancient Egypt. It is also a common shape in soccer ball designs, which are based on truncated icosahedrons.

Related polygons

The pentagon is part of a family of polygons, with the pentagram (a self-intersecting pentagon) being its most famous derivative. Truncating a pentagon produces a decagon, and it is the face of the Platonic solid known as the dodecahedron. Higher-dimensional analogs include the pentachoron (a 4-simplex) and the 5-cell. In tiling, the regular pentagon does not tessellate the plane alone, but combinations with other shapes, as seen in Penrose tilings, create aperiodic patterns. The study of these related shapes extends into fields like crystallography and the geometry of fullerene molecules such as buckminsterfullerene.

Category:Polygons Category:Elementary shapes Category:5 (number)