Angle Trisection

Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836).

Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as and radians ( and , respectively), which can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as (Honsberger 1991). In addition, trisection of an arbitrary angle can be accomplished using a marked ruler (a Neusis construction) as illustrated above (Courant and Robbins 1996).

An angle can also be divided into three (or any whole number) of equal parts using the quadratrix of Hippias or trisectrix.

An approximate trisection is described by Steinhaus (Wazewski 1945; Peterson 1983; Steinhaus 1999, p. 7). To construct this approximation of an angle having measure , first bisect and then trisect chord (left figure above). The desired approximation is then angle having measure (right figure above). To connect with , use the law of sines on triangles and gives

(1)

so . Since we also have , this can be written

(2)

Solving for then gives

(3)

This approximation is with of even for angles as large as , as illustrated above and summarized in the following table (Petersen 1983), where angles are measured in degrees.

()	()	()	()
10	3.333333	3.333804	3.332393
20	6.666666	6.670437	6.659126
30	10.000000	10.012765	9.974470
40	13.333333	13.363727	13.272545
50	16.666667	16.726374	16.547252
60	20.000000	20.103909	19.792181
70	23.333333	23.499737	23.000526
80	26.666667	26.917511	26.164978
90	30.000000	30.361193	29.277613
99	33.000000	33.486234	32.027533

has Maclaurin series

(4)

(Sloane's A158599 and A158600), which is readily seen to a very good approximation to .
Wolfram