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a regular polygon, the data generally given are its kind, and the length of a side, or a given circle within which it is to be inscribed. The ordinary solution in such cases involves the remembering of certain specific constructions which are liable to be forgotten when most needed. All that is absolutely required to be known for the con- struction of any regular polygon, is the relative position of any two of its adjacent sides, and in certain cases the length of one of them. The relative position, or, in other words, the angles made by any two adjacent sides of a regular polygon, are easily determined. The 32 FIRST PRINCIPLES OF exterior angle, or that formed by one side with the other produced, is always equal to 360 divided by the number of the sides of the polygon, and the interior angle, or that formed by the meeting of the two adjacent sides, is 180 minus the exterior angle. The angle at the centre (or central angle) of a regular polygon is equal to the exterior angle. With these simple facts committed to memory, the student or apprentice can, with a scale of chords now generally found on all pocket rules, lay down at once on his work any regular polygon having either an odd or an even number of sides. To apply these facts we will take Problem 19 (Fig. 67). To construct a regular pentagon with a given length of side. Here 360 + 5 equals 72, the exterior angle; and 180 ^ 72 = 108, the interior. Let AB be the given side, produce it (say to the left) at A, draw the line AC, making an angle of 72 with AB produced, and of a length equal to AB ; bisect AB and AC by perpendiculars intersecting in S, then S is the centre of the circumscribing circle.