Math 2.0

Geometry Foundations and axiom systems Problem

Squaring the circle

Builds on Straightedge and compass construction

Problem

Given a circle, construct with straightedge and compass alone a square with the same area.

Remarks

For a circle of radius 11, the square has side π\sqrt{\pi}. So the problem asks whether π\sqrt{\pi} is a constructible number. The answer is no: squaring the circle is impossible.

History

Anaxagoras is the first known Greek to study the problem. Hippocrates of Chios squared some figures bounded by circular arcs, the lunes of Hippocrates. In 1882, Ferdinand von Lindemann proved that π\pi is transcendental, and so that the construction is impossible.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph
Straightedge and compass construction, concept, no evidence levelStraightedge and comp…conceptSquaring the circle, problem, no evidence levelSquaring the circleproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Straightedge and compass construction concept
  2. Squaring the circle problem

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Squaring_the_circle, lead section
  2. Web https://en.wikipedia.org/wiki/Squaring_the_circle, History
About this page
Tag0BSXRER81B
URIm2://0BSXRER81B
Aliasesquadrature-of-the-circle
Kindproblem
WritingW1 stated
MSC51M15
StatusThis copy of the site has no build data. Run m2 status 0BSXRER81B.
UsesStraightedge and compass construction
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