A tetradecagon, also called a 14-gon, is a polygon with fourteen edges and fourteen vertices. The term is built from Greek roots: tetra (four) + deka (ten) + gon (angle). A tetradecagon may be convex or nonconvex; a regular tetradecagon has all sides equal and all interior angles equal and is often the primary subject in elementary geometry.

Basic measures and formulas

For a regular tetradecagon the central angle equals 2π/14 = π/7 radians (about 25.714°) and each interior angle equals (n-2)π/n = 6π/7 radians, or exactly 154 2/7 degrees. If the side length is a, the circumradius R and inradius r follow R = a/(2 sin(π/14)) and r = a/(2 tan(π/14)). The area can be written as A = (14/4) a^2 cot(π/14) = (7/2) a^2 cot(π/14), which is the standard polygon area formula specialized to n = 14.

The symmetry group of a regular tetradecagon is the dihedral group D14 of order 28, consisting of 14 rotations (including the identity) and 14 reflections. As with other even-sided regular polygons, half of the reflection axes pass through opposite vertices and the other half through midpoints of opposite edges. Subgroups include cyclic groups of rotations and smaller dihedral groups.

Star forms, compounds and constructibility

Beyond the convex regular form, 14-gons admit star polygons and compounds. Proper star polygons arise when vertices are joined by steps that are coprime to 14, producing figures commonly denoted {14/k} such as {14/3} and {14/5}. When the step shares a common divisor with 14 the result is a compound of smaller polygons (for example {14/2} decomposes into two heptagons). A regular tetradecagon is not constructible with straightedge and compass alone because 14 = 2·7 and 7 is not a Fermat prime; hence exact classical construction is not possible by the usual criterion.

History, use and notable facts

Polygons with many sides have appeared in mathematical problems, architectural ornaments and design elements where circular shapes are approximated by many-sided figures. The regular tetradecagon does not tile the plane by congruent copies because its interior angle (154 2/7°) does not divide 360° evenly; nevertheless, 14-gons may appear within more complex tilings and decorative patterns. They also serve as examples in algebraic investigations of symmetry, group actions and cyclotomic polynomials.

Examples and further reading

Concrete instances of tetradecagons include any fourteen-sided floor plan, decorative medallions, or polygonal approximations used in computer graphics to model roundness with a fixed vertex count. For concise definitions and additional illustrations see this external resource on polygons, which provides context for the 14-gon among other regular and star polygons.