POLYGON, in fortification, denotes the figure of a town, or other fortress. The exterior or external polygon is bounded by lines drawn from the point of each bastion to the points of the adjacent bas tions ; and the interior polygon is formed by lines joining the centres of the bas tions.
Poo moss, prob!nno concerning. 1. On a regular polygon to circumscribe a cir cle, or to circumscribe a regular polygon upon a circle ; bisect two of the angles of the given polygon, A and B, (fig. 15), by the right lines, A F, B F ; and on the point, F, where they meet, with the radius, A F, describe a circle, which will circumscribe the polygon. Next, to cir cumscribe a polygon, divide 360 by the number of sides required, to find e d; which set off from the centre, F, and draw the line, de, on which construct the polygon as in the following problem. 2. On a given line to describe given regular polygon : find the angle of the polygon in the table, and in E set off an angle equal thereto ; then drawing E A =E D through the points E, A, ll, de scribe a circle, and in this applying the given right line as often as you can, the polygon will be described. 3. To find
the sum of all the angles in any given regular polygon: multiply the number of sides by 18U° ; from the product sub tract 366°, and the remainder is the sum required : thus, in a pentagon, 180 X 5 = 900, and 900 — 360 = 540, the sum of all the angles in a pentagon. 4. To find the area of a regular polygon : mul tiply one side of the polygon by half the number of sides ; and then multiply this product by a perpendicular, let fall from the centre of the circumscribing circle, and the product will be the area re quired: thus, if A B (the side of a pen. tagon) =54X 23 = 135, and 135 x 29 (the perpendicular) = 3915 = the area required. 5. To find the area of an ir regular polygon, let it be resolved into triangles, and the sum of the areas of these will be the area of the polygon.