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the rest of the joke: What is round is also square and what is square is also round. The round square and the square round have exactly the same volume.
Squaring the circle is sometimes used as an idiom in common discourse to mean taking on an impossible task.
Squaring the circle is sometimes used as an idiom in common discourse to mean taking on an impossible task.
==Joke about the section of a cylinder as a squared circle==
===The section of a cylinder as a squared circle===
As a joke, a circle can simultaneously present a square if a perfectly round [[cylinder]] is produced with a sectional length equal to its measured diameter. Looked at from the side it presents a square configuration. Looked at from the end it presents a circular configuration. The circle coexists as a square object and the square coexists as a circular object. The circle has been squared and the square has been circled, by the shaping of one cylinder. No one takes this seriously as being related to the classical problem.
As a mathematical joke, a circle can simultaneously present a square if a perfectly round, solid [[cylinder]] is produced with a sectional length equal to its measured diameter.
The straightedge determines the exactitude of the first straight smooth cut across the round cylinder at a right angle to the straight body of the cylinder (1). The compass point is placed at what is determined to be the exact center of the diameter of the round crosscut end of the solid cylinder, and the stylus is adjusted to the outer edge of the cut and rotated around the edge to determine with exactitude the center of the crosscut and the length of the radius of the cut (2). The span of the compass is then fixed and it is then moved 90<sup>o</sup> to the straight side of the cylinder and positioned with the stylus at the exact edge of the crosscut, with the point of the compass positioned exactly at a 90<sup>o</sup> angle from the edge of the cut (3). The stylus of the compass is then rotated 180<sup>o</sup> away from the edge of the first crosscut end to determine with exactitude where to make the second right angle cut across the cylinder (4). The second straightedge smooth crosscut of the cylinder makes a finished cylinder section with a measured exact length identical to the measured exact diameter of the cylinder, a cylindrical section exactly as long as it is wide (5).
Looked at from the side it presents a square configuration. Looked at from the end it presents a circular configuration. The perfect circle formed here by crosscut exists as an exactly squared object and the exact square formed here by the same crosscut (twice) exists as a perfectly circular object. The circle has been squared and the square has been circled. What is perfectly round is also exactly square and what is exactly square is also perfectly round. The perfectly round exact square and the exactly square perfect round together are both in essence one object having the same single surface area ''in toto''. This one square and round object has identically the same single volume as itself, in itself, by itself, by the exactly squared shaping of one perfectly round cylinder.
If the external surface of the straight side of the cylinder only is opened up and laid flat, its area can be determined by measuring the four sides of the resultant rectangle with a straightedge. The lengths of the top edge and the bottom edge only of the opened and flattened area of the straight sides of the cylinder are identical to the circumference of the circular ends, each length taken and divided by four and rearranged as four equal straight sides of a square, the area of the resultant square determined for one end of the cylinder is then multiplied by two for both ends, and the resulting combined area of the two ends added to the area of the opened and flattened straight side of the cylinder, give the area of the squared-off circular surface of the single cylindrical section with both round ends, the area of this one single squared circle object determined by straightedge measurement (6).
No one takes this seriously as being properly related in any way to an actual solution of the classical problem posed by the ancient geometers, as a real solution to an insoluble problem in six simple steps.
See [[Paradox]].
==External links==
==External links==