trisecting the angle

geometry

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July 14, 2026

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Trisecting an angle is a classical problem in geometry that involves dividing a given angle into three equal parts. This task was historically attempted using only a straightedge (an unmarked ruler) and a compass, tools that Euclid mandated for geometric constructions. However, it was proven in the 19th century that trisecting an arbitrary angle is impossible using only these tools.1

Despite this impossibility, ancient mathematicians devised methods using other means. For instance, the ancient Egyptians likely encountered the problem when dividing time at night, and it's probable that Greek geometers faced similar challenges related to angular measurement.1

Methods developed included using a mechanical device called a quadratrix, invented by Hippias of Elis, and a technique known as neusis, which involves maneuvering a measured length into a specific position. Archimedes, for example, used a method of neusis with a marked straightedge to achieve angle trisection.12

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Trisecting the Angle
Archimedes’ Method
Euclid’s insistence (c. 300 bc) on using only unmarked straightedge and compass for geometric constructions did not inhibit the imagination of his successors. Archimedes (c. 285–212/211 bc) made use of neusis (the sliding and maneuvering of a measured length, or marked straightedge) to solve one…
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Euclid’s insistence (c. 300 bc) on using only unmarked straightedge and compass for geometric constructions did not inhibit the imagination of his successors. Archimedes (c. 285–212/211 bc) made use of neusis (the sliding and maneuvering of a measured length, or marked straightedge) to solve one…
Read More

mathematics
The three classical problems
The early history of angle trisection is obscure. Presumably, it was attempted in the pre-Euclidean period, although solutions are known only from the 3rd century or later.
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The early history of angle trisection is obscure. Presumably, it was attempted in the pre-Euclidean period, although solutions are known only from the 3rd century or later.
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Trisecting the Angle
The Quadratrix of Hippias
Hippias of Elis (fl. 5th century bc) imagined a mechanical device to divide arbitrary angles into various proportions. His device depends on a curve, now known as the quadratrix of Hippias, that is produced by plotting the intersection of two moving line segments, as shown…
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Hippias of Elis (fl. 5th century bc) imagined a mechanical device to divide arbitrary angles into various proportions. His device depends on a curve, now known as the quadratrix of Hippias, that is produced by plotting the intersection of two moving line segments, as shown…
Read More