trisection of angle

Given an angle of measure ( α such that 0<απ2, one can construct an angle of measure α3 using a compass and a ruler ( with one mark on it as follows:

  1. 1.

    Construct a circle c with the vertex ( O of the angle as its center. Label the intersectionsMathworldPlanetmathPlanetmath of this circle with the rays of the angle as A and B. Mark the length OB on the ruler.

  2. 2.

    Draw the ray AO.

  3. 3.

    Use the marked ruler to determine Cc and DAO such that CD=OB and B, C, and D are collinearMathworldPlanetmath. Draw the line segmentMathworldPlanetmath BD¯. Then the angle measure of CDO is α3. (The line segment OC¯ is drawn in red. Having this line segment drawn is useful for reference purposes for the justification of the construction.)


Let m denote the measure of an angle. Then this construction is justified by the following:

  • Since AOB is an exterior angleMathworldPlanetmath of BOD, we have that m(AOB)=m(OBD)+m(ODB);

  • Since OC=OB=CD, we have that BOC and OCD are isosceles trianglesMathworldPlanetmath;

  • Since the angles of an isosceles triangle are congruentPlanetmathPlanetmath, m(OBC)=m(OCB) and m(COD)=m(CDO);

  • Since OCB is an exterior angle of OCD, we have that m(OCB)=m(COD)+m(CDO);

  • Note that OBC=OBD and ODB=CDO;

  • Thus,


Note that, since angles of measure π6, π3, and π2 are constructible using compass and straightedge, this procedure can be extended to trisect any angle of measure β such that 0<β2π:

  • If 0<βπ2, then use the construction given above.

  • If π2<βπ, then trisect an angle of measure β-π2 and add on an angle of measure π6 to the result.

  • If π<β3π2, then trisect an angle of measure β-π and add on an angle of measure π3 to the result.

  • If 3π2<β2π, then trisect an angle of measure β-3π2 and add on an angle of measure π2 to the result.

This construction is attributed to Archimedes.


  • 1 Rotman, Joseph J. A First Course in Abstract Algebra. Upper Saddle River, NJ: Prentice-Hall, 1996.
Title trisection of angle
Canonical name TrisectionOfAngle
Date of creation 2013-03-22 17:16:35
Last modified on 2013-03-22 17:16:35
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Algorithm
Classification msc 01A20
Classification msc 51M15
Related topic VariantsOnCompassAndStraightedgeConstructions