We certainly can square the circle with the right devices.
- Make the circle the base of a cylinder.
- Mark off spot on a string.
- Wrap the string around the cylinder along the edge of its base and mark off a second spot one lap from the original mark.
- Sraighten the string so now the two marks are separated by the circumference.
- Using compass and straightedge methods, construct a rectangle whose dimensions are half the circumference and the radius, and square this rectangle.
Archimedes is better known for approximating the circumference rather than rendering an exact construction:
The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant". Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that
223/71 < π < 22/7 (that is 3.1408 < π < 3.1429).
Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 5 June 2013. English translation by Catriona and David Lischka, p. 169.
Ibid., p. 170.
Ibid., pp. 175, 205.