ON THE CONSTRUCTION OF REGULAR 



POLYGONS. 



By H. STANLEY REDGROVE, B.Sc. (Lond.), F.C.S., and W. H. COLES. 



There are simple well-known methods for con- 

 structing regular figures of 3, 4, 5, 6, and 8 sides 

 of given length,* but the construction of regular 

 polygons with 7 or more than 8 sides, the length 

 of the sides being given, is a more difficult matter. 

 There is a well-known method (which need not be 

 described here) involving the division of a semicircle 

 into as many equal parts as the polygon required is 

 to have sides, which can be used with accuracy when 

 the number of sides is either 2" or 3 x2" (i.e., 4, 8, 

 16, 32, and so on, or 6, 12, 24, 48, and so on), since 

 the bisection of any arc and the trisection of a 

 semicircle are both easily effected. When the 

 number of sides, however, is any other number (e.g., 

 7, 9, 10, 11, and so on), this method suffers under 

 the disadvantage that the semicircle has to be 

 divided by trial. 



Another method is to calculate the vertical angle 

 (d) of the polygon by means of the formula 



x 

 where x is the number of sides ; or the central 

 angle ((j>) may be calculated from the formula 



_ 360° , 



x 



either of these angles is then set out. In many 

 cases, however, these angles are not measured by an 

 exact number of degrees, and cannot, therefore, be 

 accurately set out with an ordinary protractor. 

 Moreover, when x is large, an increase of 1 or 

 more to x produces very little alteration in the size 

 of either 6 or <p, so that the angles would have to be 

 measured with extreme accuracy for the method to 

 be any good. (See Table 58.) 



The following method, though not new, does not 

 appear to be very generally known. Let A B (see 

 Figure 363) be one side of the required polygon. 

 Bisect A B in M, and at M draw M P perpendicular 

 to A B and of indefinite length. With centre M 

 and radius M A (or M B) describe an arc cutting 

 M P in 4. This point obviously gives the centre of 

 the circle circumscribing the square drawn on A B 

 as base. With centre B and radius B A describe an 

 arc cutting M P in 6. This point obviously gives 

 the centre of the circle circumscribing the regular 



hexagon drawn on A B as base. Bisect 46 in 5 and, 

 commencing from 6, mark off divisions 7, 8, 9, 10, and 

 so on, along 6 P, each equal to 45 or 56. Then each 

 of these points gives, approximately, the centre of 

 the circle circumscribing the regular polygon, drawn 

 on A B as base, containing the same number of sides 

 as the number used to designate the point. It is a 

 simple matter, then, to draw the circle circumscrib- 

 ing the polygon desired, and to step distances equal 

 to A B around its circumference. In Figure 363 

 are shown polygons containing 5, 7, 9, and 12 sides 

 thus drawn. The first three are fairly accurate. 

 The dodecagon, however, is not satisfactory, and 

 the method is increasingly inaccurate as the number 

 of sides is increased ; for example, if one attempts 

 to draw a fifteen-sided figure by this method, the 

 result is a polygon with sixteen sides. 



In the present paper we propose giving the results 

 of an investigation of this method, which has made 

 it possible to devise three similar, but far more 

 accurate, methods, whereby regular polygons con- 

 taining as many as twenty sides can be satisfactorily 

 and easily drawn. In the above method it is assumed 

 that the distances of the centres of the circumscribing 

 circles from the mid-point of the given side are 

 in arithmetical progression. Now, the length of 

 this distance, in any case, measured in terms of 

 half the side as unit, is obviously given by the 

 tangent of half the vertical angle of the polygon, 



/ 180°\ 

 i.e., tan ( 90° J, where x is the number of 



■j n. ir ,„ 6M , / Qno 180°\ 



sides; thus in Figure 363, , „ = tan 90 — — ^— I 



AM V 6 / 



= tan 60° = 1-732 (approx.). This method 



180°^ 



assumes, therefore, that tan ( 90° 



is a linear 



function of x. This is not actually the case, since 

 rf tan(90"- 1 -f) .see* (90° -if) 



dx * a 



i.e., a quantity whose value is not constant. This 



can also be seen from the figures given in Table 59. 



/ 180°\ 

 In column 2 are given the values of tan (90° — ) 



between x = It and .* = 20. In column 3 are 



* There are also special methods for a few other polygons, but they are all tedious. 



t No figures, of course, can be drawn corresponding to x — 1 and x = 1, but the values corresponding thereto are 



given throughout the table for the sake of completeness. 



337 



