When a polygon has SIX SIDES, it is called a HEXAGON. To draw a Regular Hexagon of which A B is one side, Fig. 45. On A B draw an equilateral triangle, A B C. Produce the sides of the equilateral triangle beyond the point C until they cut the perpendiculars in D and E. Draw the horizontal line D E, and you will then have another equilateral triangle, DE C, upside down—that is, its apex touching the apex of the former one. Now through C draw a horizontal line, and on it, from the centre, mark off the length of the line. CD-namely, C F and C G. Draw the lines D F, F A, and E G, G B, which will complete the regular hexagon. You will observe, then, that the Regular Hexagon consists of six equilateral triangles. You will remember that I told you, on page 41 that four right angles would meet at a point, and completely fill up the space around it; and so will six Equilateral Triangles, or three Hexagons, for you will see that each of the angles of a hexagon is double that of an equilateral triangle, as two equilateral triangles meet there. To describe a Circle around a Hexagon. This could be done by drawing the circle to touch all the six angles of the hexagon; but to obtain more points, draw lines through the centre, and through the middle of the sides. Mark off on these lines, from the centre, the length of C D (or any other of the lines which reach from the centre to the angles). This will give H, I, J, K, L, M, and now you will have twelve points through which you can trace your circle. The above is a design for a floor-tile or other ornament for covering a flat surface, which is based upon the hexagon. To draw this ornament Firstly, draw the hexagon A B C DEF. Draw lines from A to C, from C to F, and from A to F, and you will have an equilateral triangle inscribed in the Hexagon. Then draw the lines B D, D E, and E B, which will form another triangle crossing the former one, and a perfect hexagon will be formed in the middle, having a small equilateral triangle on each of its sides, and when lines are drawn from the angles to the centre, the central hexagon will be divided into six equilateral triangles equal to those on its outer sides; the whole star, therefore, is made up of twelve equilateral triangles. Next draw the double lines in the design, and shade either like the copy or in any way you think it will look best. A polygon having seven sides is called a Heptagon. To draw a heptagon on the given side A B, Fig. 47. Draw a perpendicular at B, and the quadrant A C. Divide this quadrant into seven equal parts. You can only do this by trying, as there is not any rule by which you could get the seven divisions; but a little practice will soon enable you to get over the difficulty, and you will remember having complete the heptagon, and you will see that the figure will consist of seven triangles equal to А О В. A circle may be described about the heptagon by producing OH and O I, and similar lines drawn in each of the triangles until they are equal in length to the lines O A or O B, and these, together with the seven angles of the pentagon, will give fourteen points through which the circle may be drawn. To inscribe a circle in the heptagon, draw lines similar to OH and O I; in each of the triangles mark the length of these on the lines O A, O B, &c., and you will thus have fourteen points through which the circle may be drawn within the heptagon. A polygon with eight sides is called an Octagon. To draw an octagon on the given side A B (Fig. 48). Draw perpendiculars at A and B. Produce the line A B at each end, and you will thus have a right angle outside A, and another outside B. Bisect both these right angles by the lines A E and B F, which must be equal in length to A B. Draw a line from E to F, cutting the perpendicu |