POLYGON, in geometry, a figure with many sides, or whose perimeter consists of more than four sides at least : such are the pentagon, hexagon, heptagon, &c.
Every polygon may be divided into as many triangles as it has sides ; for if you assume a point, as a, (see Plate XII. Miscel. fig. 14), any Where within the polygon, and from thence draw lines to every angle, a b, a c, a d, &c. they shall make as many triangles as the figure has sides. Thus, if the polygon 'lath six sides (as in the figure above) the double of that is twelve, from whence take four, and there remains eight : I say, that all the angles, b, c, d, e, j; g, of that poly gon, taken together, are equal to eight right angles. For the polygon having six sides, is divided 'into six triangles ; and the three angles of each by 1.32 Bud. are equal to two right ones ; so that all the angles together make twelve right but each of these triangles bath one angle in the point, a, and by it they complete the space round the same point ; and all the angles about, a point are known to be equal to four right ones, wherefore those four taken from twelve, leave eight, the sum of the right angles of the hexagon. So it is plain the figure bath twice as many right angles as it hath sides, except four.
Every polygon circumscribed about a circle, is equal to a rectangled-triangle, one of whose legs shall be the radius of the circle, and the other the perimeter (or sum of all the sides) of the polygon. Hence, every regular polygon is equal to a rectangled-triangle, one of whose legs is the perimeter of the polygon, and the other a perpendicular drawn from the centre to one of the sides of the polygon.
And every polygon circumscribed about a circle is bigger than it ; and every polygon inscribed is less than the circle, as is manifest, because the thing con taining is always greater than the thing contained. The perimeter of every poly gon circumscribed about a circle, is greater than the circumference of that circle, and the perimeter of every poly gon inscribed is less. Hence, a circle is equal to a right-angled triangle, whose base is the circumference of the circle, and its height the radius of it.
For this triangle will be less than any polygon circumscribed, and greater than any inscribed ; because the circumfer ence of the circle, which is the base of the triangle, is greater than the compass of any inscribed, therefore it will be equal to the circle. For, if this triangle be greater than any thing that is less than the circle, and less than any thing that is greater than the circle, it follows, that it must be equal to the circle. This is called the quadrature, or squaring of the circle ; that is, to find a right-lined figure equal to a circle, upon a supposition that the basis given is equal to the circum ference of the circle ; but actually to find a right line equal to the circumference of a circle, is not yet discovered geome trically.