The argument below shows that it is impossible to construct a 20В° angle.

The angle ПЂ / 3 radians (60 degrees, written 60В°) is constructible. Therefore, any number that is constructible by a sequence of steps is a root of a minimal polynomial whose degree is a power of two. Every irrational number that is constructible in a single step from some given numbers is a root of a polynomial of degree 2 with coefficients in the field generated by these numbers. This equivalence reduces the original geometric problem to a purely algebraic problem.Д®very rational number is constructible. It follows that, given a segment that is defined to have unit length, the problem of angle trisection is equivalent to constructing a segment whose length is the root of a cubic polynomial. The triple-angle formula gives an expression relating the cosines of the original angle and its trisection: cos ОёВ = 4 cos 3 Оё / 3 в€’ 3 cos Оё / 3. From a solution to one of these two problems, one may pass to a solution of the other by a compass and straightedge construction. The problem of constructing an angle of a given measure Оё is equivalent to constructing two segments such that the ratio of their length is cos Оё. However, Wantzel published these results earlier than Г‰variste Galois (whose work, written in 1830, was published only in 1846) and did not use the concepts introduced by Galois. Wantzel's proof, restated in modern terminology, uses the concept of field extensions, a topic now typically combined with Galois theory. Pierre Wantzel published a proof of the impossibility of classically trisecting an arbitrary angle in 1837. These 'solutions' often involve mistaken interpretations of the rules, or are simply incorrect. Other techniques were developed by mathematicians over the centuries.Д«ecause it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. It is possible to trisect an arbitrary angle by using tools other than straightedge and compass.

For example, it is relatively straightforward to trisect a right angle (that is, to construct an angle of measure 30 degrees). However, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. Pierre Wantzel proved in 1837 that the problem, as stated, is impossible to solve for arbitrary angles. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. The example shows trisection of any angle Оё> 3ПЂ / 4 by a ruler with length equal to the radius of the circle, giving trisected angle П†= Оё / 3.ДЄngle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics.

Angles may be trisected via a neusis construction using tools beyond an unmarked straightedge and a compass.

Angle ruler