TOPICS

# 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
 (4)

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

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

## Explore with Wolfram|Alpha

More things to try:

## References

Bogomolny, A. "Angle Trisection." http://www.cut-the-knot.org/pythagoras/archi.shtml.Bogomolny, A. "Angle Trisection by Hippocrates." http://www.cut-the-knot.org/Curriculum/Geometry/Hippocrates.html.Bold, B. "The Problem of Trisecting an Angle." Ch. 5 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 33-37, 1982.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190-191, 1996.Courant, R. and Robbins, H. "Trisecting the Angle." §3.3.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 137-138, 1996.Coxeter, H. S. M. "Angle Trisection." §2.2 in Introduction to Geometry, 2nd ed. New York: Wiley, p. 28, 1969.Dixon, R. Mathographics. New York: Dover, pp. 50-51, 1991.Dörrie, H. "Trisection of an Angle." §36 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 172-177, 1965.Dudley, U. The Trisectors. Washington, DC: Math. Assoc. Amer., 1994.Geometry Center. "Angle Trisection." http://www.geom.umn.edu:80/docs/forum/angtri/.Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 25-26, 1991.Klein, F. "The Delian Problem and the Trisection of the Angle." Ch. 2 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 13-15, 1980.Loy, J. "Trisection of an Angle." http://www.jimloy.com/geometry/trisect.htm.Ogilvy, C. S. "Solution to Problem E 1153." Amer. Math. Monthly 62, 584, 1955. Ogilvy, C. S. "Angle Trisection." Excursions in Geometry. New York: Dover, pp. 135-141, 1990.Peterson, G. "Approximation to an Angle Trisection." Two-Year Coll. Math. J. 14, 166-167, 1983.Scudder, H. T. "How to Trisect and Angle with a Carpenter's Square." Amer. Math. Monthly 35, 250-251, 1928.Sloane, N. J. A. Sequences A158599 and A158600 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wantzel, M. L. "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas." J. Math. pures appliq. 1, 366-372, 1836.Wazewski, T. Ann. Soc. Polonaise Math. 18, 164, 1945.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 25, 1991.Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971.

## Cite this as:

Weisstein, Eric W. "Angle Trisection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AngleTrisection.html