decagon

A decagon is a polygon with ten sides and ten vertices. In general, any ten-sided polygon is called a decagon, while a regular decagon has all sides equal and all interior angles equal. The sum of interior angles in a decagon is (n − 2) × 180°, which for n = 10 equals 1440°, so each interior angle of a regular decagon is 1440°/10 = 144°. The exterior angle is 36°, and a regular decagon exhibits dihedral symmetry of order 20 (the group D10), including ten rotational symmetries and ten mirror symmetries.

For a regular decagon, various measurements are expressed in terms of the circumradius R (the radius of the circumscribed circle). The side length is s = 2R sin(π/10) = 2R sin 18° ≈ 0.6180R, and the inradius is r = R cos(π/10) = R cos 18° ≈ 0.9511R. The area can be written as A = (1/2) n R^2 sin(2π/n) = 5R^2 sin 36° ≈ 2.939R^2, or in terms of the side length as A = (n s^2)/(4 tan(π/n)) = 10s^2/(4 tan 18°) ≈ 7.692s^2. The decagon has 35 diagonals, connecting non-adjacent vertices; these diagonals come in several lengths corresponding to central angles of 72°, 108°, 144°, and 180° (the latter being opposite vertices).

In tiling and design, decagons appear in various contexts. They cannot tile the plane by themselves in a periodic pattern, but decagonal motifs occur in aperiodic tilings such as Penrose tilings, and decagons are used in some architectural and ornamental designs.