Let a b (Figs. 386, 387, and 388) be given lines, equal to a side of the required figure. From b draw bc at right angles to a b; upon a and b, with a b for radius, describe the arcs acd and feb; divide ac into as many equal parts as the polygon is to have sides, and extend those divisions from c towards d; from the second point of division, counting from c towards a, as 3 (Fig. 386), 4 (Fig. 387), and 5 (Fig. 388), draw a line to b; take the distance from said point of division to a, and set it from b to e; join e and a; upon the intersection o with the radius oa, describe the circle afdb; then radiating lines, drawn from b through the even numbers on the arc a d, will cut the circle at the several angles of the required figure.
In the hexagon (Fig. 387), the divisions on the arc ad are not necessary; for the point o is at the intersection of the arcs a d and fb, the points f and d are determined by the intersection of those arcs with the circle, and the points above g and h can be found by drawing lines from a and b through the centre 0. In polygons of a greater number of sides than the hexagon the intersection o comes above the arcs; in such case, therefore, the lines ae and b5 (Fig. 388) have to be extended before they will intersect. This method of describing polygons is founded on correct principles, and is therefore accurate. In the circle equal arcs subtend equal angles (Arts. 357 and 515). Although this method is accurate, yet polygons may be described as accurately and more simply in the following manner. It will be observed that much of the process in this method is for the purpose of ascertaining the centre of a circle that will circumscribe the proposed polygon. By reference to the Table of Polygons in Art. 442 it will be seen how this centre may be obtained arithmetically. This is the rule: multiply the given side by the tabular radius for polygons of a like number of sides with the proposed figure, and the product will be the radius of the required circumscribing circle. Divide this circle into as many equal parts as the polygon is to have sides, connect the points of division by straight lines, and the figure is complete. For example: It is desired to describe a polygon of 7 sides, and 20 inches a side. The tabular radius is 1.15238. This multiplied by 20, the product, 23.0476 is the required radius in inches. The Rules for the Reduction of Decimals, in the Appendix, show how to change decimals to the fractions of a foot or an inch. From this, 23.0476 is equal to 23 1/16 inches, nearly. It is not needed to take all the decimals in the table, three or four of them will give a result sufficiently near for all ordinary practice.
Let a, b and c (Fig. 389) be the given lines. Draw the line de and make it equal c; upon e, with b for radius, describe an arc at f; upon d, with a for radius, describe an arc intersecting the other at f; join d and f, also f and e; then dfe will be the triangle required.
Let abed (Fig. 390) be the given figure. Make ef (Fig. 391) equal to cd; upon f, with da for radius, describe an arc at g; upon e, with ca for radius, describe an arc intersecting the other at g; join g and e, upon f and g, with db and ab for radius, describe arcs intersecting at h; join g and h, also h and f; then Fig. 391 will every way equal Fig. 390.
So, right-lined figures of any number of sides may be copied, by first dividing them into triangles, and then proceeding as above. The shape of the floor of any room, or of any piece of land, etc., may be accurately laid out by this problem, at a scale upon paper; and the contents in square feet be ascertained by the next.