Decagon

A decagon is a polygon with ten sides and ten interior angles.^[1] The term originates from the Greek words deka meaning "ten" and gōnia meaning "angle," reflecting its structure as a ten-angled figure.^[2] Decagons can be classified as regular or irregular based on the equality of their sides and angles. A regular decagon has all sides of equal length and all interior angles measuring exactly 144 degrees, while an irregular decagon may have varying side lengths and angles.^[3] The sum of the interior angles in any decagon is 1440 degrees, calculated using the formula (n-2) \times 180^\circ where n=10.^[4] Each exterior angle of a regular decagon measures 36 degrees.^[3] In geometry, decagons appear in constructions involving circles and are notable for their symmetry; a regular decagon can be inscribed in a circle, with vertices equally spaced at intervals of 36 degrees.^[5] They have historical significance in ancient mathematics, such as in Euclidean geometry where constructions of regular decagons relate to the golden ratio through pentagonal relationships.^[6]

Definition and classification

Definition

A decagon is a polygon with exactly ten sides and ten vertices.^[7]^[8] As a closed, two-dimensional figure, it encloses a region bounded by these straight line segments, each connecting consecutive vertices.^[9] The term "decagon" originates from the Ancient Greek words déka (δέκα), meaning "ten," and gonía (γωνία), meaning "angle" or "corner," reflecting its ten-angled structure.^[2]^[10] This nomenclature first appeared in ancient Greek geometry, where mathematicians like Euclid explored properties of regular polygons, including the decagon, in works such as Elements around 300 BCE.^[11] For any simple decagon, the sum of its interior angles is (10-2) \times 180^\circ = 1440^\circ.^[8]^[12] In the specific case of a regular decagon, where all sides and angles are equal, each interior angle measures $144^\circ, while each exterior angle is $360^\circ / 10 = 36^\circ.^[7]^[12]

Types of decagons

Decagons, as ten-sided polygons, can be classified based on their geometric properties such as convexity, regularity, and intersection characteristics. These classifications help distinguish various forms beyond the basic definition of a closed figure with ten straight sides. The primary categories include convex and concave decagons, regular and irregular variants, as well as simple and complex types.^[13]^[7] Convex decagons are those in which all interior angles measure less than 180 degrees, ensuring that the polygon lies entirely on one side of each of its sides with no indentations or intersecting sides. This configuration allows all diagonals to lie within the interior of the polygon, making convex decagons suitable for many geometric applications where uniformity and enclosure are required. In contrast, concave or re-entrant decagons feature at least one interior angle greater than 180 degrees, creating an indentation or "dent" that causes part of the polygon to bend inward, though the sides do not intersect themselves. These shapes can resemble star-like forms without self-intersection, providing flexibility in modeling non-convex boundaries in design and architecture.^[8]^[14]^[7] Regular decagons represent the most symmetric type, characterized by all sides of equal length (equilateral) and all interior angles equal (equiangular), resulting in a highly uniform structure with rotational symmetry of order 10. This regularity ensures that the decagon is both convex and simple, serving as a foundational shape in classical geometry. Irregular decagons, on the other hand, deviate from this uniformity by having unequal side lengths or unequal interior angles, yet they still sum to 1440 degrees for the interior angles. Within irregular decagons, subtypes include equilateral decagons, where all sides are equal but angles vary, allowing for non-regular shapes like rhombi extended to ten sides; isogonal decagons, which have equal angles and are vertex-transitive under symmetry, often being cyclic but not necessarily equilateral; and isotoxal decagons, which possess equal edge lengths and equivalent angles in a generalized symmetric sense, frequently appearing in dual relationships to isogonal forms.^[13]^[15]^[16]^[17]^[18] Decagons are further distinguished as simple or complex based on self-intersection. Simple decagons maintain a boundary that does not cross itself, encompassing both convex and concave varieties where the edges form a single closed loop without overlaps. Complex decagons, by contrast, have sides that intersect each other, creating intricate patterns such as star polygons, which violate the non-intersecting rule of simple polygons and often exhibit higher degrees of symmetry in their crossings. This distinction is crucial for understanding topological properties and applications in tiling or artistic designs.^[13]^[14]^[19]

Regular decagon

Construction

The regular decagon is constructible using a compass and straightedge, as its number of sides n = 10 = 2 \times 5 factors into a power of 2 and the distinct Fermat prime 5, satisfying the necessary conditions for classical constructibility established by Gauss.^[20] This allows the division of a circle into 10 equal 36° arcs through geometric operations. One standard method begins with the construction of a regular pentagon inscribed in a circle, leveraging the pentagon's fivefold rotational symmetry, which is intrinsically linked to the golden ratio.^[21] To obtain the decagon, first locate the center of the circle by drawing the bisectors of two non-adjacent interior angles of the pentagon, which intersect at the center. Next, draw radii from the center to each vertex of the pentagon, confirming the central angles of 72°. Then, bisect each of these 72° central angles using the standard compass-and-straightedge angle bisection: from the center, draw a small arc intersecting the two radii, then from those intersection points draw equal arcs to find a point on the bisector, and draw the line from the center through that point to intersect the circle. The 10 points on the circle—alternating between the original pentagon vertices and the new bisection points—form the vertices of the regular decagon. Connecting these points sequentially yields the polygon. This process ensures all sides and angles are equal due to the congruence of the isosceles triangles formed by the radii and bisected arcs.^[22] An alternative approach uses intersecting circles to mark the 36° intervals, based on the construction of points related to the golden ratio in pentagonal geometry. This method starts with a circle and diameters, constructing additional circles at midpoints of radii to find key intersection points that locate vertices at 36° increments around the circle. Details of this construction tie directly to the fivefold symmetry and are analogous to steps in regular pentagon construction.^[23]

Dimensions and formulas

The dimensions of a regular decagon are interrelated through its side length s, circumradius R (the radius of the circumscribed circle), and apothem a (the inradius, or distance from the center to a side). The side length s is given by the formula s = 2R \sin\left(\frac{\pi}{10}\right), which follows from the central angle of \frac{2\pi}{10} = 36^\circ subtended by each side in the circumscribed circle.^[24] Equivalently, solving for the circumradius yields R = \frac{s}{2 \sin\left(\frac{\pi}{10}\right)}.^[25] The apothem a represents the perpendicular distance from the center to any side and is expressed as a = R \cos\left(\frac{\pi}{10}\right).^[24] In terms of the side length, this becomes a = \frac{s}{2} \cot\left(\frac{\pi}{10}\right).^[25] These relations derive from basic trigonometry in the isosceles triangle formed by two radii and one side of the decagon. The exact values of the trigonometric functions involved are \sin\left(\frac{\pi}{10}\right) = \frac{\sqrt{5} - 1}{4} and \cos\left(\frac{\pi}{10}\right) = \frac{\sqrt{10 + 2\sqrt{5}}}{4}, which can be derived using half-angle formulas from the known values for \frac{\pi}{5}.^[26] These expressions simplify computations for the decagon's dimensions and highlight connections to the golden ratio, as detailed in the section on mathematical relations. Substituting them yields closed-form expressions such as R = \frac{s}{2} \csc\left(\frac{\pi}{10}\right) = \frac{1 + \sqrt{5}}{2} s.^[25] Chord lengths in the circumscribed circle correspond to arcs spanning different numbers of sides and are calculated using the general formula for a regular polygon: the chord subtending a central angle of \frac{2k\pi}{10} (for k = 1, 2, \dots, 5) has length $2R \sin\left(\frac{k\pi}{10}\right).^[24] For example, the side length is the chord for k=1, while longer chords represent diagonals; the chord for k=5 equals the diameter $2R. These lengths provide the distances between vertices separated by k steps around the decagon.^[24]

Area

The area of a regular decagon can be calculated by dividing the polygon into 10 congruent isosceles triangles, each formed by two radii to adjacent vertices and the side between them. The central angle of each triangle is $36^\circ, so the area of one such triangle is \frac{1}{2} r^2 \sin 36^\circ, where r is the circumradius. The total area is therefore the sum of these 10 triangles:
A = 10 \cdot \frac{1}{2} r^2 \sin 36^\circ = 5 r^2 \sin 36^\circ.
The exact value of \sin 36^\circ is \frac{\sqrt{10 - 2 \sqrt{5}}}{4}, yielding
A = \frac{5}{4} r^2 \sqrt{10 - 2 \sqrt{5}}.
This expression provides an approximate numerical value of A \approx 2.938 r^2.^[27] Equivalently, the area can be expressed in terms of the side length s. Using the general formula for the area of a regular polygon, A = \frac{1}{4} n s^2 \cot \frac{\pi}{n} with n = 10, simplifies to
A = \frac{5}{2} s^2 \sqrt{5 + 2 \sqrt{5}}.
This yields an approximate value of A \approx 7.694 s^2. The presence of the shared radical term \sqrt{5 + 2 \sqrt{5}} in the decagon's area formula and that of the regular pentagon reflects their common symmetries.^[25]^[28]

Mathematical relations

Golden ratio

The golden ratio, denoted \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618, is fundamentally connected to the proportions of the regular decagon.^[21] In a regular decagon with side length s, the ratio of the circumradius r to the side is given by \frac{r}{s} = \frac{1}{2 \sin(\pi/10)}, where \sin(\pi/10) = \sin 18^\circ = \frac{\sqrt{5} - 1}{4}. This simplifies to \frac{r}{s} = \phi, since \frac{\sqrt{5} - 1}{4} = \frac{1}{2\phi}. An equivalent radical form for related trigonometric quantities, such as \cos 18^\circ = \frac{\sqrt{10 + 2\sqrt{5}}}{4}, underscores how \phi emerges from the decagon's angular structure.^[25]^[26] Certain diagonals in the regular decagon exhibit ratios involving \phi relative to the side length; specifically, the diagonal spanning three vertices has length \phi^2 s, reflecting the iterative nature of golden ratio proportions in polygonal geometry.^[29] The diagonals of a regular decagon intersect to form regular pentagons at various scales, embedding \phi within nested structures where the ratio of each pentagon's diagonal to its side is precisely \phi.^[30] A key trigonometric identity linking the decagon to \phi is \cos 36^\circ = \frac{\phi}{2}, derived from the central angle subtended by three sides and confirmed through geometric constructions involving isosceles triangles with vertex angle $36^\circ.^[31] These \phi-based relations also appear in the construction of the decagon and its area formulas, providing elegant simplifications for exact computations.^[25]

Symmetry

The symmetry group of the regular decagon is the dihedral group D_{10}, which consists of 20 elements: 10 rotations and 10 reflections that map the decagon onto itself.^[5]^[32] The rotational symmetries are generated by a counterclockwise rotation of $36^\circ about the center, yielding 10 distinct rotations by angles k \times 36^\circ for k = 0, 1, \dots, 9.^[5]^[32] The reflectional symmetries occur across 10 axes passing through the center: 5 axes each connecting a pair of opposite vertices, and 5 axes each connecting the midpoints of a pair of opposite sides.^[5] The subgroup consisting solely of the rotations is isomorphic to the cyclic group \mathbb{Z}_{10}. Additionally, the decagon exhibits pentagonal symmetry through dihedral subgroups isomorphic to D_5, such as the one generated by a $72^\circ rotation and an appropriate reflection.^[32]^[33] Visually, the 10 axes of reflection intersect at the center, partitioning the plane into 10 congruent $36^\circ sectors. A fundamental domain for the action of D_{10} is any one of these sectors—bounded by portions of two adjacent reflection axes—such that repeated application of the group elements generates the full symmetry of the decagon.^[5]^[32]

Variants and extensions

Dissections

A regular decagon can be dissected into triangles through triangulation, a process that divides any simple convex n-gon into n-2 non-overlapping triangles using non-intersecting diagonals, yielding the minimum of 8 triangles for a decagon.^[34] This method preserves the area and is fundamental in computational geometry for decomposing polygons.^[35] By connecting the center of a regular decagon to its 10 vertices, it divides into 10 congruent isosceles triangles, each with two sides equal to the radius and a base equal to the side length of the decagon.^[25] These triangles share the central vertex and exploit the rotational symmetry of the figure. Using diagonals, a regular decagon can be further dissected into regular pentagons and related shapes; for instance, it can be cut into two regular pentagons and two pentagrams using six pieces.^[36] Such decompositions often leverage the diagonals' lengths, which are multiples of the golden ratio relative to the side length, to align pieces precisely.^[25] In the context of Hilbert's third problem, which originally addressed equidecomposability of polyhedra, the two-dimensional counterpart—the Wallace–Bolyai–Gerwien theorem—establishes that a decagon can be dissected into finitely many polygonal pieces to form any other simple polygon of equal area, demonstrating area equivalence through reassembly.^[37] A specific dissection utilizing golden ratio diagonals involves dividing the decagon to isolate a central pentagon surrounded by smaller decagonal regions, where the scaling factors follow golden ratio proportions derived from intersecting diagonals.^[38] The dihedral symmetry of the decagon guides these dissection lines to ensure balanced and congruent components.

Skew decagon

A skew decagon is a 10-sided skew polygon, meaning its 10 vertices do not all lie in one plane, resulting in a non-planar figure embedded in three-dimensional space. Unlike planar decagons, the edges of a skew decagon alternate between different parallel or non-parallel planes, often forming a zigzag path that traverses multiple layers. This configuration arises naturally in the study of polyhedra, where such polygons serve as equatorial belts or helical paths.^[39] The properties of a skew decagon include exactly 10 edges connecting its vertices in a closed loop, but without being confined to a single plane, its interior region is not well-defined in the traditional two-dimensional sense. Concepts like density—measuring how the polygon overlaps itself in projection—and winding number, which quantifies the number of rotational turns around an axis, can be extended to skew decagons to analyze their topological behavior in 3D embeddings, though these are more complex than for planar polygons due to the spatial twisting. For instance, in orthogonal projections along certain axes, a skew decagon may appear as a regular decagon, highlighting its underlying symmetry.^[40] In uniform polyhedra, skew decagons appear as zigzag paths, notably in the regular dodecahedron, where six such polygons function as Petrie polygons—closed sequences of edges where every two consecutive edges share a face, but no three do. These Petrie skew decagons wind around the dodecahedron, alternating vertices across parallel planes separated by the polyhedron's equatorial belts. The petrial dodecahedron, a regular skew polyhedron and the Petrie dual of the regular dodecahedron, further exemplifies this by having six skew decagons as its faces, demonstrating their role in uniform compounds and Archimedean-like structures. Petrie polygons represent a subtype of skew decagons in this context.^[41]

Star decagons

Star decagons, also known as decagrams, are non-convex, self-intersecting regular star polygons with ten vertices. They are denoted using Schläfli symbols of the form {10/n}, where n is an integer coprime to 10 or leading to compounds, specifically for n=3 and n=4, as {10/2} reduces to the pentagon {5} and {10/5} to the digon {2}. These figures share the same vertices as a regular decagon but connect them in a skipping pattern to form stars.^[42] The {10/3} decagram is a single-component star polygon with density 3, meaning its edges wind around the center three times before closing. It is constructed by placing ten equally spaced points on a circle and connecting every third point, resulting in a ten-pointed star. Alternatively, it can be formed by drawing two concentric regular pentagons rotated relative to each other by 36 degrees, where the edges of the pentagons intersect to outline the star. The intersection points of its edges form an inner regular pentagon, and various segment ratios within the figure, such as certain side-to-diagonal lengths, incorporate the golden ratio φ ≈ 1.618.^[42] The {10/4} decagram, with density 4, is not a single connected component but a compound consisting of two interlocked regular pentagrams {5/2}. It is constructed similarly by connecting every fourth vertex among the ten points on a circle, but due to the greatest common divisor of 10 and 4 being 2, it decomposes into the two separate pentagrams rotated by 36 degrees relative to each other. This compound exhibits the same ten-fold rotational symmetry as the regular decagon and shares its vertices.^[42]