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 pi/2 and pi radians (90 degrees and 180 degrees, respectively), which can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as 3pi/7 (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 A having measure alpha, first bisect A and then trisect chord BE (left figure above). The desired approximation is then angle DAB having measure t (right figure above). To connect t with alpha/3, use the law of sines on triangles DeltaDAB and DeltaEAD gives


so sint=2sinbeta. Since we also have beta=(alpha/2)-t, this can be written


Solving for t then gives


This approximation is with 1 degrees of alpha/3 even for angles alpha as large as 120 degrees, as illustrated above and summarized in the following table (Petersen 1983), where angles are measured in degrees.

alpha ( degrees)alpha/3 ( degrees)t ( degrees)s ( degrees)

t has Maclaurin series


(OEIS A158599 and A158600), which is readily seen to a very good approximation to alpha/3.

See also

Angle Bisector, Archimedes' Spiral, Circle Squaring, Conchoid of Nicomedes, Cube Duplication, Cycloid of Ceva, Maclaurin Trisectrix, Morley's Theorem, Neusis Construction, Origami, Pierpont Prime, Quadratrix of Hippias, Tomahawk, Trisectrix

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Cite this as:

Weisstein, Eric W. "Angle Trisection." From MathWorld--A Wolfram Web Resource.

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