POLYGON (Gr. rolls, many; geinia, a corner), a plane figure, bounded by a number of straight lines; the name is conventionally limited to those plane figures whose bounding straight lines are more than four in number. Polygons of 5, 6, 7, 8, etc., sides are denominated pentagons, hexagons, heptagons, octagons, etc.; and when the number of sides exceeds twelve, the figure is merely mentioned as a polygon of so many sides. The quindecagon, or figure of 15 sides, is the only common exception to this rule. Polygons have many general properties; such as that the sum of the angles of a polygon, when increased by four right angles, or 360', is equal to twice as many right angles as there are sides in the polygon, and that (supposing the number of sides of the polygon to be expressed by n) the number of its diagonals is v(n — 3). alsoif apolygon of an even number of sides he circumscribed about a circle, the sums of its even and odd sides are equal; and if a polygon of an even number of sides be inscribed in a circle, the sums of its even and odd angles are equal. A polygon which has till its sides and angles equal
is called a regular polygon. All polygons of this class are capable of ben. ;nscrilnd in or circumscribed about a circle; but though the problem is merely to divide the circum ference of a circle into a number of equal parts, corresponding to the number of sides in the polygon, geometry was till lately only able to perform it in those cases where the number of sides of the polygon belongs to one or other of the series 2, 4, 8. 16, etc.: 3, 6, 12, 24. etc.; or 5, 10, 20, 40, etc. Gauss (q.v.), however, in the beginning of the present century, showed how it could be done in the case of all polygons, the number of whose sides was of the form 2" + 1 (provided it be a prime number), or a multiple of this prime number by any power of 2. This discovery supplies its with innumerable series representative of the numbers of the sides of polygons which can be described around or inscribed in a circle, such as 17, 34, 68, etc.; 257, 514, 1028, etc.