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MECHANICAL AND ENGINEERING DKAWlNG A. ' ftp $9. 34 MECHANICAL AND ENGINEERING DRAWING Problem 21 (Fig. 69). To construct a reyular octayon, with a given lenyth of side. Here 360 -f 8 = 45; and 180 - 45 - 135. Let AB be the given side. Produce it in both directions, and at A and B draw lines AE, BF, of the same length as AB, and making with AB produced angles of 45 ; bisect the angles formed at A and B, and their inter- section at S will be the centre of the circumscribing circle. With SA or SB as radius, describe this circle, step AB round it from E to F in the points 1, 2, 3, 4; join E 1, 2, 3, 4 F, and the required octagon is constructed. To inscribe an octagon in a given circle : Draw two radii (Fig. 69) at an angle of 45 to each other, and they will cut the circle in points A and B ; join AB, and it will be a side of the octagon. Its length stept round the circle will give the same points as in the previous construction ; join them, and an octagon will be inscribed in the given circle. 15. The same principle of construction as used in the last three problems is applicable to any regular polygon, whatever may be the number of its sides ; but in practice it is preferable to subdivide the sides of those we have given if the division will give the required number of sides than to lay down an independent construction, the chances of not obtaining the exact length of the side of the polygon required increasing as the number of sides increase. On paper, and