43. Upon a Given Side to Draw a Regular Undec-agon. - In Fig. 172, A B represents the given side upon which a regular undecagon is to be drawn. Produce A B indefinitely in the direction of D. From B as center, and with B A as radius, draw the semicircle A M D. Through the point B, perpendicular to A B, draw the line H D indefinitely. From B as renter, and with B F as radius, strike the arc F G, cutting the perpendicular H G in the point G. From G as center, and G D as radius, strike the arc D H, cutting the perpendicular H G in the point H. With D as center, and D H as radius, strike the arc H M, cutting the semicircle in the point M. Draw M D, which bisect, obtaining the point K, through which, from B, draw the line B K, and produce it until it cuts the semicircle in the point E. Then B E will be another side of the required figure. Bisect the two sides now obtained and erect perpendicular lines, producing them until they intersect, as shown by F C and L C. Then C, the point of intersection, is the center of the circle which circumscribes the uudecagon. From C as center, and with C A as radius, strike the circle, as shown. Set the dividers to the space A B and step off the circumference, obtaining the points O, V, T, R, P, S, N and I. Draw the chords A O, O V, V T, T R, R P, P S, S N, N I and I E, thus completing the figure.

Fig. 173. - Upon a Given Side to Draw a Regular Dodecagon.

44. Upon a Given Side to Draw a Regular Dodecagon. - In Fig. 173, let A B represent the given side upon which a regular dodecagon is to be drawn. Produce A B indefinitely in the direction of D. From B as center, and with B A as radius, describe the semicircle A F D. From D as center, and with D B as radius, describe the are B F, cutting the semicircle in the point F. Draw F D, which bisect by the line V B, cutting the semicircle in the point E. Then E B is another side of the dodecagon. From the middle points of the two sides now obtained, as G and H, civet perpendiculars, as shown, cutting each other at the point, C. This point of intersection, C, then is the center of the circle which will circumscribe the required dodecagon. From C as center, and with CB as radius, strike the circle, as shown. Set the dividers to the distance A B and space off the circumference, thus obtaining the points L, P, M, S, N, B, O, K and 1. Draw the connecting lines L P, P M. M S, S N, N R, R O, O K, K I and I E, thus completing the figure.

45. General Rule by which to Draw any Regular Polygon, the Length of a Side Being Given. - With a radius equal to the given side describe a semicircle, the circumference of which divide into as many equal parts as the figure is to have sides. From the center by which the semicircle was struck draw a line to the second division in the circumference. This line will be one side of the required figure, and one-half of the diameter of the semicircle will be another, and the two will be in proper relationship to each other. Therefore, bisect each, and through their centers erect per pendiculars, which produce until they intersect. The point of intersection will be the center of the circle which will circumscribe the polygon. Draw the circle, and setting the dividers to the length of one of the sides already found, step off the circumference, thus obtaining points by which to draw the remaining sides of the figure.

Fig. 174. - Upon a Given Side to Construct a Regular Polygon of Thirteen Sides by the General Rule.

46. To Construct a Regular Polygon of Thirteen Sides by the General Rule, the Length of a Side being Given. In Fig. 174, let A B be the given side. With B as center, and with B A as radius, describe the semicircle A F G. Divide the circumference of the semicircle into thirteen equal parts, as shown by the small figures, 1, 2, 3, 4, etc. From B draw a line to the second division in the circumference, as shown by B 2. Then A B and B 2 are two of the sides of the required figure, and are in correct relationship to each other. Bisect A B and B 2, as shown, and draw DC and E C through their central points, prolonging them until they intersect at the point C. Then C is the center of the circle which will circumscribe the required polygon. Strike the circle, as shown. Set the dividers to the space A B, and step off corresponding spaces in the circumference of the circle, as shown, and connect the several points so obtained by lines, thus completing the figure. 47. Within a Given Square to Draw a Regular Octagon. - In Fig. 175, let A D B E be any given square within which it is required to draw an octagon. Draw the diagonals D E and A B, intersecting at the point C. From A, D, B and E as centers, and with radius equal to one-half of one of the diagonals, as A C, strike the several arcs H N, G K, I M and L O, cutting the sides of the square, as shown. Connect the points thus obtained in the sides of the square by drawing the lines G O, H I, K L and M N, thus completing the figure.

For general use a very convenient scale may be constructed, as shown in Fig. 176, from which half the length of one side of a polygon of any number of sides and of any diameter in inches and fractions of inches may readily be obtained. Draw the vertical line O B and divide it into inches and parts of an inch. From these points of division draw horizontal lines; from the point O draw the following lines and at the following angles from the horizontal line O P:

Fig. 175. - Within a Given Square to Draw a Regular Octagon.

 A line at 75° for polygons having 12 sides. ,, 72° ,, ,, 10 ,, ,, 67 1/2° ,, ,, 8 ,,
 A line at 60° for polygons having 6 sides. ,, 54° ,, ,, 5 ,, ,, 45° ,, ,, 4 ,,

The figures on O B will designate the radius of the inscribed circle measured from O. The distance from O B on any horizontal line to the oblique line denoting the required polygon will be half the length of a side of the polygon of the diameter indicated by the figure at the end of the horizontal line assumed. The distance from O measured upon the oblique line to the assumed horizontal line will be the radius of the circumscribed circle.

Fig. 176 - Scale for Constructing Polygons of any Number of Sides, the Diameter of the Inscribed Circle Being Given in Inches. - Half Full Size.

In the engraving three polygons are drawn showing the application of the scale.