Squaring the Circle

A Math Forum Project

Table of Contents:

Famous Problems Home

The Bridges of Konigsberg
· Euler's Solution
· Solution, problem 1
· Solution, problem 2
· Solution, problem 4
· Solution, problem 5

The Value of Pi
· A Chronological Table of Values
· Squaring the Circle

Prime Numbers
· Finding Prime Numbers

Famous Paradoxes
· Zeno's Paradox
· Cantor's Infinities
· Cantor's Infinities, Page 2

The Problem of Points
· Pascal's Generalization
· Summary and Problems
· Solution, Problem 1
· Solution, Problem 2

Proof of the Pythagorean Theorem

Proof that e is Irrational

Book Reviews



Squaring the circle is one of the three great problems of Classical Geometry, along with the trisection of the angle and the duplication of the cube. Since 1800 B.C. mathematicians have worked on the problem of constructing a square equal in area to that of a given circle. Whether or not this is possible depends, of course, on what tools you allow yourself. Plato insisted that the problem be solved with straightedge and compass only. To achieve this requires constructing a length equal to sqrt(pi) times the radius of the circle. Thus when Lindemann proved in 1882 that Pi is transcendental (not the root of any polynomial with rational coefficients) he effectively proved that the construction was impossible with only straightedge and compass.

Please enjoy this picture while the Java Applet is loading.

Click on point C and drag to change the area of the square. Can you make the area approximately equal to that of the circle? What is the length of a side of the square when the areas are equal? How does this compare to the length of the radius of the circle?

Sorry, this page requires a Java-compatible web browser.

This applet was created using a prototype of JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum Press, Inc. All rights reserved. Portions of this work are being funded by the National Science Foundation (awards DMI 9561674 & 9623018).

to A Chronological Table of Values Attributed to Pi
to Prime Numbers

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August, 1998