CHAPTER XVIII. Regular Polygons (Continued) .. 1. To inscribe a regular hexagon in a circle: A hexagon having six sides, the angle subtended at the centre of the circle by the side of a regular hexagon inscribed in the circle, will be of 360° = 60° Using then the protractor, or adjusting the bevel to an angle of 60', lay off at the centre two angles of 60°. Produce the three sides of these angles both ways to the circumference, and join the succeeding points of intersection. The construction being accurately made, the bevel will show the equality of the angles ABC, BCD, and the dividers will show the equality of the sides AB, BC, Since, however, each of the triangles in the figure is equilateral, having its sides equal to the radius, the sides of the hexagon are equal to the radius of the circle. Hence the easiest way to describe a hexagon in a circle is to measure off, with the dividers, six chords in succession, each equal to the radius. Evidently the angle of a regular hexagon is 120°. 2. If tangents to the circle the angular points of the hexagon ABCDEF, the tangents form another hexa- L gon, which is said to be about the circle. The equality D of the sides GH, HK, .... may be tested with the dividers, and K the equality of the angles GHK, HKL, ... with the bevel. B H 3. If we wish to construct a regular hexagon with sides of given length, we describe a circle with radius of this length, and in it inscribe a regular hexagon as in $ 1. A E 4. To inscribe a regular octagon in a circle : We may construct at the centre eight angles, each of 45°, and join the ends of consecutive radii bounding these angles ; angles; or, perhaps more conveniently, we may proceed as follows': Draw two diameters at right angles to one another and join their extremities. We thus have a square in the circle. Through the centre, using parallel rulers, draw diameters parallel to the sides of uare. The quadrants are thus bisected, and we get eight equal angles at the centre. Joining ends of the successive radii which bound these angles, we have an octagon inscribed in the circle. The accuracy the square. of the construction may be tested by using the dividers to determine whether the sides are equal, and the bevel to determine whether the angles are equal. Each of the angles at the centre is 45°. Hence each of the angles at the base of any of the isosceles triangles, OAB, OBC, ... is 674", and the angle of a ° regular octagon is 135o. 5. If tangents be drawn at the angular points of the octagon ABCDEFGH, the tangents form another regular octagon which is said to be about the circle. 6. To describe a regular octagon with side, AB, of given length we may proceed as follows: Construct the angle ABC of 135°, and make BC = AB. Bisect AB and BC in K and L, and draw KO, LO perpendicular to AB and BC. With O as centre, and radius 0A, OB or OC describe a circle. On this lay off with the dividers six chords equal to AB or BC, beginning at the point C or A. That the rest of the circle is exactly taken up with six such chords affords a test of the accuracy with which the angle ABC (135°) is constructed, AB and BC are bisected, and the perpendiculars KO and LO are drawn. 7. The pupil may continue these exercises, constructing regular decagons, dodecagons, etc., in a way quite analogous to the preceding constructions. A B The radius of a circle being 1{ in., inscribe in it a regular hexagon. Test the accuracy of your construction by testing the equality of all the angles. Describe a regular hexagon about the circle in the preceding question, testing the equality of sides and angles of the figure. Construct a regular hexagon with sides 11 in. Construct a figure similar to that annexed, in which the outer circle touches six smaller ones. Construct the figure also SO that the six small circles touch one another, and are all touched by the outer (large) and inner (small) circles. (Radius of small circles should be one-third radius of large circle.) Describe a regular octagon in a circle whose radius is 43 millimetres. Test the accuracy of your construction by testing the equality of the sides (using dividers), and by examining whether each of the angles of the octagon is 135o. Construct a regular octagon whose side is 2 inches. Examine the accuracy of your construction by testing, with the dividers, the equality of the sides, and, with the bevel, the equality of the angles. Describe eight circles of the same radius, each touching two others of the set, and the entire eight lying within and being touched by a ninth circle of given radius. The general way of solving such a problem as the be 8, 9, preceding is as follows: Suppose the number of small circles is to Let AOB be the 8th, 9th, as the case may be, part of 360°. Bisect the angles OAB, OBA by AC, BC. Through C draw DCE parallel to AB. Then evidently DA, DC, EB, EC are all equal, and the circle described with D as centre, and DA or DC as radius, will touch the circle described with E as centre, and EB or EC as radius; and both circles will touch the large one. B Exercises. 1. In a circle of radius 13 in., inscribe a regular hexagon. 3. Describe a regular hexagon with side of 2 in. Join alternate angles, so obtaining a star-shaped figure with six points. What is the six-sided figure at centre of this? Apply tests. What are the various triangles in the figure? Apply tests. 4. In the figure of the preceding question, at what various angles are the sides of the hexagon at centre inclined to any side of the original hexagon? 5. About a circle of radius 40 millimetres describe a hexagon with angles 90°, 100°, 110, 130°, 140°, 150°. 6. A regular hexagon is described about a circle of radius 2 in. Show that the side of the hexagon is *3 in. 7. The side of a regular hexagon is 2 in. What is the length of the radius of the circle inscribed in it ? 8. Inscribe a regular octagon in a circle of radius 32 millimetres. Test accuracy of construction. 9. In a circle of radius 50 millimetres, inscribe a regular octagon, ABCDEFGH. Join AD, DG, GB, ...., each time passing over two angles, and so obtaining a star-shaped figure with eight points. What is the figure formed at centre? Apply tests. |