31. To Inscribe a Regular Undecagon within a Given

Circle. - In Fig. 101, let B D A L be any given circle in which a regular figure of eleven sides is to be drawn. Draw any diameter, as B A. and draw a radius, as D C. at right angles to B A. Bisect C A. thus obtaining the point E. From E as center, and with F D as radius, describe the arc D F, cutting B A in the point F. With D as center, and D F as radius, describe the are F G, cutting the circumference in the point G. Draw the chord G D and bisect it, as shown by H O, thus obtaining the point K. From D as center, and with D K as radios, cut the circumference in the point I. Draw I D. Then I D will be equal to one side of the required figure. Set the dividers to this space and step off the points in the circumference, as shown by N, R, S. M, P, L, O, T and V. and draw the connecting chords, as shown, thus completing the figure.

32. To Inscribe a Regular Dodecagon within a Given Circle. - In Fig. 162, let M F A I he any given circle in which a dodecagon is to be drawn. From any point in the circumference, as A. with a radius equal to the radius of the circle, describe the arc C B, cutting the circumference in the point B. Draw the chord A B, which Insect as shown, and draw the line O C, cutting the circumference in the point D. Draw A D, which will then he one side of the given figure. With the dividers set to this space step off in the circumference the points B, I, N, H, M, G, L, F, K and E, and draw the several chords, as shown, thus com pleting the figure.

33. General Rule for Inscribing any Regular Polygon in a Given Circle. - Through the given circle draw any diameter. At right angles to this diameter draw a radius. Divide that radius into four equal parts, and prolong it outside the circle to a distance equal to three of those parts. Divide the diameter of the circle into the same number of equal parte as the polygon is to have sides. Then from the end of the radius prolonged, as above described, through the second division in the diameter, draw a line cutting the circumference. Connect this point, in the circumference and the nearest end of the diameter. The line thus drawn will be one side of the required figure. Set the dividers to this space and step off on the circumference of the circle the remaining number of sides and draw connecting lines, which will complete the figure.

Fig. 161. - To Inscribe a Regular Un-decagon within a Given Circle.

Fig. 162. - To Inscribe a Regular Dodecagon within a Given Circle.

Fig. 163. - To Inscribe a Regular Undec-agon within a Given Circle by the General Rule.

Fig. 164. - Upon a Given side to Construct an Equilateral Triangle.

Fig. 165. - To Construct a Triangle, the Length of the Three Sides being Given.

Fig. 166. - Upon a Given Side to draw a Regular Pentagon.

34. To Inscribe a Regular Polygon of Eleven Sides (Undecagon) within a Given Circle by the General Rule.

- Through the given circle, F D F G in Fig. 163, draw any diameter, as B F. which divide into the same number of equal parts as the figure is to have sides, as shown by the small figures. At right angles to the diameter just drawn draw the radius D K, which divide into four equal parts. Prolong the radius D K outside the circle to the extent of three of those parts, as shown by a b c, thus obtaining the point c. From C, through the second division in the diameter, draw the line c H, cutting the circumference in the point H. Connect H and E. Then H E will be one Bide of the required figure. Set the dividers to the distance H F and step off the circumference, as shown, thus obtaining the points for the other sides, and draw the connecting arcs, all as illustrated in the figure.

35. Upon a Given Side to Construct an Equilateral Triangle. - In Fig. 164, let A B represent the length of the given side. Draw any line, as C D, making it equal to A B. Take the length A B in the dividers, and placing one foot upon the point C, describe the arc F F. Then from D as center, with the same radius, describe the are G H. intersecting the first arc in the point K. Draw K C and K D. Then C D K will be the required triangle.

36. To Construct a Triangle, the Length of the Three

Sides being Given. - In Fig. 165, let A B, C D and E

F be the given sides from which it is required to construct a triangle. Draw any straight line, G H, making it in length equal to one of the sides, E F. Take the length of one of the other sides, as A B, in the compasses, and from one end of the line just drawn, as G, for center describe an arc, as indicated by L M. Then set the compasses to the length of third side, C D, and from the opposite end of the line first drawn, H, describe a second arc, as I K, intersecting the first in the point O. Connect O G and O H. Then O G H will be the required triangle.

37. Upon a Given Side to Draw a Regular Pentagon. - In Fig. 166, let A B represent the given side upon which a regular pentagon is to be constructed. With B as center and B A as radius, draw the semicircle A D E. Produce A B to E. Bisect the given side A D, as shown at the point F, and erect a perpendicular, as shown by F C. Also erect a perpendicular at the point B, as shown by G H. With B as center, and F B as radius, strike the arc F G, cutting the perpendicular H G in the point G. Draw G E. Willi G as center, and G E as radius, strike the arc E H, cutting the perpendicular in the point H. With E as center, and E H as radius, strike the arc H D, cutting the semicircle A D E in the point D. Draw D B, which will be the second side of the pentagon. Bisect D B, as shown, at the point K, and erect a perpendicular, which produce until it intersects the perpendicular F C, erected upon the center of the given side in point F. Then C is the center of the circle which circumscribes the required pentagon. From C as center, and with C B as radius, strike the circle, as shown. Set the dividers to the distance A B and step off the circumference of teh circle, obtaining the points M and L. Draw A M, M L and L D, which will complete the figure.

Fig. 167. - Upon a Given Side to Draw a Regular Hexagon.

Fig. 168. - Upon a Given Side to Draw a Regular Heptagon.

Fig. 169. - Upon a Given Side to Draw a Regular Octagon.