Angle trisection is the construction of an angle one-third as large as a given angle. Some angles can be trisected by straightedge and compass construction, but there is no such construction for every angle.
For a concrete obstruction, trisecting would construct . Set . The triple-angle identity givesThe only possible rational roots of this monic cubic are , and neither is a root. The cubic is therefore irreducible over , giving degree three. A constructible number has degree a power of two, so this angle cannot be constructed from a unit length and the given constructible angle using those tools.
In contrast, can be trisected by constructing from an equilateral triangle. The failure concerns a general construction procedure; it does not prohibit all individual trisections, numerical approximations, or constructions with other tools.
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Angle trisection is the problem of dividing an arbitrary angle into three equal parts using only a compass and straightedge, which is one of the classical problems of ancient Greek geometry. The problem can be traced back to the works of ancient mathematicians, and it remains significant in the history of mathematics because it was proven to be impossible to accomplish using only these traditional tools for any general angle.