squaring the circle

geometry

Also known as: quadrature of the circle

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 14, 2026

Quick Summary

"Squaring the circle" refers to the ancient geometric problem of constructing a square with the same area as a given circle using only a straightedge and compass. This task has been proven impossible.123

The impossibility stems from the nature of pi ((\pi)), which is a transcendental number. Ferdinand von Lindemann proved in 1880 that squaring the circle is impossible because (\pi) cannot be constructed using only a straightedge and compass.3

While the full circle cannot be squared, ancient mathematicians like Hippocrates of Chios demonstrated that certain moon-shaped areas called lunes could be squared, meaning their area could be expressed as a rectilinear area. This led to some hope that the circle itself might be manageable, but ultimately, the problem remained unsolvable with the allowed tools.43

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

4 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

Quadrature of the Lune
…that someone had accomplished the squaring of the circle. Whether the unknown genius used lunes or some other method is not known, since for lack of space Boethius did not give the demonstration. He thus transmitted the challenge of the quadrature of the circle together with fragments of geometry apparently…
Read More
…that someone had accomplished the squaring of the circle. Whether the unknown genius used lunes or some other method is not known, since for lack of space Boethius did not give the demonstration. He thus transmitted the challenge of the quadrature of the circle together with fragments of geometry apparently…
Read More

mathematics
The three classical problems
…cube), trisecting the angle, and squaring the circle. Even in the pre-Euclidean period the effort to construct a square equal in area to a given circle had begun. Some related results came from Hippocrates (see Sidebar: Quadrature of the Lune); others were reported from Antiphon and Bryson; and Euclid’s theorem…
Read More
…cube), trisecting the angle, and squaring the circle. Even in the pre-Euclidean period the effort to construct a square equal in area to a given circle had begun. Some related results came from Hippocrates (see Sidebar: Quadrature of the Lune); others were reported from Antiphon and Bryson; and Euclid’s theorem…
Read More

Hippocrates of Chios
In his attempts to square the circle, Hippocrates was able to find the areas of certain lunes, or crescent-shaped figures contained between two intersecting circles. He based this work upon the theorem that the areas of two circles have the same ratio as the squares of their radii. A…
Read More
In his attempts to square the circle, Hippocrates was able to find the areas of certain lunes, or crescent-shaped figures contained between two intersecting circles. He based this work upon the theorem that the areas of two circles have the same ratio as the squares of their radii. A…
Read More

Ferdinand von Lindemann
…classical Greek construction problem of squaring the circle (constructing a square with an area equal to that of a given circle) by compass and straightedge is insoluble.
Read More
…classical Greek construction problem of squaring the circle (constructing a square with an area equal to that of a given circle) by compass and straightedge is insoluble.
Read More