Tetradecagon
A tetradecagon, also known as a tetrakaidecagon or 14-gon, is a polygon consisting of fourteen sides and fourteen vertices.[1] In its regular form, all sides are of equal length and all interior angles are equal, measuring approximately 128.57 degrees each, with the sum of interior angles totaling 2160 degrees as determined by the formula (n-2)×180° for an n-sided polygon where n=14.[2][3] The regular tetradecagon possesses dihedral symmetry of order 28, denoted by the Schläfli symbol {14}, and its area can be calculated as A = \frac{14}{4} a^2 \cot \frac{\pi}{14} \approx 15.3345 a^2, where a is the side length.[2] Unlike some regular polygons, a regular tetradecagon cannot be constructed using only a compass and straightedge due to the involvement of the prime factor 7 in its vertex count, which does not satisfy the conditions for classical constructibility.[4] It appears in various geometric contexts, such as stellations and dissections, and serves as a Petrie polygon in certain higher-dimensional polytopes.[5]Definitions and Fundamentals
General tetradecagon
A tetradecagon is a polygon with exactly fourteen edges and fourteen vertices, where each edge connects two consecutive vertices to form a closed chain.[6][1] In general, it need not have equal side lengths or angles, distinguishing it from the regular variant.[7] Tetradecagons may be classified as simple or self-intersecting based on whether their edges cross one another. Simple tetradecagons form a Jordan curve with no self-intersections, while self-intersecting forms, such as certain star polygons, feature overlapping edges. Among simple tetradecagons, convex examples lie entirely on one side of each supporting line through their edges, whereas concave ones have at least one reflex interior angle exceeding 180° and indentations.[8][9] For any simple tetradecagon, the sum of its interior angles measures 2160°, obtained by triangulating the polygon into twelve non-overlapping triangles, each with an angle sum of 180°. This holds irrespective of convexity, as the triangulation covers the interior without gaps or overlaps. Self-intersecting tetradecagons lack a well-defined interior in the same topological sense, rendering the standard angle sum inapplicable without additional conventions.[10][7]Regular tetradecagon
A regular tetradecagon is a regular polygon consisting of fourteen sides of equal length and fourteen interior angles of equal measure.[2] The measure of each interior angle is given by the formula for regular polygons, \frac{(n-2) \times 180^\circ}{n}, where n=14, yielding \frac{12 \times 180^\circ}{14} = \frac{6 \times 180^\circ}{7} \approx 154.2857^\circ. In the complex plane, the vertices of a regular tetradecagon inscribed in the unit circle are located at the points e^{2\pi i k / 14} for integers k = 0, 1, \dots, 13, corresponding to angular positions of $2\pi k / 14 radians from the positive real axis.[2] The geometry of the regular tetradecagon relates to that of the regular heptagon, as $14 = 2 \times 7; the side length for a circumradius of 1 is $2 \sin(\pi / 14), which equals $2 \cos(3\pi / 7), with the value of \cos(3\pi / 7) obtainable as a root of the equation derived from the triple-angle formula or the minimal polynomial for $2\cos(2\pi / 7).[11]Geometric Properties
Angles and side relations
The interior angle of a regular tetradecagon measures \frac{(14-2) \times 180^\circ}{14} = \frac{1080^\circ}{7} \approx 154.2857^\circ.[12] [7] The corresponding exterior angle is \frac{360^\circ}{14} = \frac{180^\circ}{7} \approx 25.7143^\circ.[12] [7] The central angle subtended by each side at the center of the circumscribed circle equals the exterior angle, \frac{360^\circ}{14} \approx 25.7143^\circ.[7] For a regular tetradecagon inscribed in a unit circle (circumradius r = 1), the side length is the chord spanning one vertex interval: $2 \sin\left(\frac{\pi}{14}\right).[13] More generally, chords spanning k adjacent vertices have length $2 \sin\left(\frac{k\pi}{14}\right) for integer k = 1 to $7, where k=1 gives the side and k=7 yields the diameter $2.[14] These chord lengths establish proportional relations among sides and diagonals: the six distinct diagonal types (for k=2 to k=7) increase monotonically from $2 \sin\left(\frac{2\pi}{14}\right) = 2 \sin\left(\frac{\pi}{7}\right) to $2, with symmetry ensuring equal lengths for spans of k and $14-k vertices.[14] The longest proper diagonal (k=6) measures $2 \sin\left(\frac{6\pi}{14}\right) = 2 \sin\left(\frac{3\pi}{7}\right) \approx 1.9499, approaching but not reaching the diameter.[14]| Span k | Chord type | Length (unit circle) |
|---|---|---|
| 1 | Side | $2 \sin(\pi/14) |
| 2 | Shortest diagonal | $2 \sin(2\pi/14) |
| 3 | Diagonal | $2 \sin(3\pi/14) |
| 4 | Diagonal | $2 \sin(4\pi/14) |
| 5 | Diagonal | $2 \sin(5\pi/14) |
| 6 | Longest proper diagonal | $2 \sin(6\pi/14) |
| 7 | Diameter | $2 \sin(7\pi/14) = 2 |
Area, perimeter, and radius formulas
The perimeter P of a regular tetradecagon is P = 14s, where s denotes the side length.[15] The side length relates to the circumradius R (distance from center to vertex) via s = 2R \sin\left(\frac{\pi}{14}\right), derived by bisecting the central isosceles triangle with apex angle \frac{2\pi}{14} = \frac{\pi}{7}.[16] The inradius r (or apothem, distance from center to side midpoint) is r = R \cos\left(\frac{\pi}{14}\right), obtained from the same bisection yielding adjacent over hypotenuse in the right triangle.[16] The area A follows from summing the areas of 14 isosceles triangles with two sides R and included angle \frac{\pi}{7}, giving A = 7 R^2 \sin\left(\frac{\pi}{7}\right).[16] Equivalently, A = \frac{1}{2} P r = 7 s r, or directly in terms of side length as A = \frac{7 s^2}{2} \cot\left(\frac{\pi}{14}\right) \approx 15.3345 s^2.[15]Diagonals and chord lengths
In a regular tetradecagon inscribed in a circle of radius R, the chord lengths between vertices separated by k steps along the perimeter are given by d_k = 2R \sin\left(\frac{k\pi}{14}\right) for k = 1, 2, \dots, 7, where k=1 yields the side length and k=2 to k=7 yield the diagonals, with k=7 being the diameter $2R. These lengths arise from the central angle \frac{2k\pi}{14} = \frac{k\pi}{7} subtended by the arc, using the standard relation in the isosceles triangle formed by two radii and the chord. The squared lengths simplify to d_k^2 = 2R^2 \left(1 - \cos\left(\frac{k\pi}{7}\right)\right), linking them algebraically to the cosines \cos\left(\frac{m\pi}{7}\right) for m=1,2,3, as higher k values reflect symmetries \sin\left(\frac{(14-k)\pi}{14}\right) = \cos\left(\frac{k\pi}{7}\right). The values \cos\left(\frac{\pi}{7}\right), \cos\left(\frac{3\pi}{7}\right), and \cos\left(\frac{5\pi}{7}\right) are the roots of the irreducible cubic equation $8x^3 - 4x^2 - 4x + 1 = 0 over the rationals, derived from the triple-angle formula \cos(3\theta) = 4\cos^3\theta - 3\cos\theta applied to \theta = \frac{\pi}{7}, \frac{3\pi}{7}, \frac{5\pi}{7}, yielding a degree-3 minimal polynomial consistent with the real subfield of the 7th cyclotomic extension. Thus, the diagonal lengths for k=2 to k=6 are algebraic numbers of degree at most 3 (or 6 in the full cyclotomic field when expressed via sines), while ratios such as \frac{d_2}{d_1} = \frac{\sin(\pi/7)}{\sin(\pi/14)} reduce via angle identities to expressions solvable from the same cubic. The diameter d_7 = 2R is rational, distinguishing it as the sole exact diagonal in closed radical form without invoking the polynomial. For R=1, the approximate diagonal lengths (to four decimal places) are as follows, computed from the trigonometric values tied to the cubic roots:| k (steps) | Diagonal type | Length d_k \approx |
|---|---|---|
| 2 | Shortest diagonal | 0.8678 |
| 3 | 1.2470 | |
| 4 | 1.5637 | |
| 5 | 1.8019 | |
| 6 | Near-diameter | 1.9499 |
| 7 | Diameter | 2.0000 |