Construction of a Square with Area Equal to that of a Given
Most of us learn the Pythagorean Theorem as a formula: . However, the ancient Greeks thought of it geometrically, and visually: "The area of the square constructed over the hypotenuse of a right triangle is equal to the sum of the areas of the squares constructed over the other two sides." The Greeks' concept led them to realize that for any given plane figure whose sides were straight lines, they could construct a square of equal area. First, they would divide the figure into triangles. Next, for each of those triangles, they would construct a square with the same area. (See http://www.geogebratube.org/material/show/id/13133.) Finally, they would add the areas of the squares via the Pythagorean Theorem, as shown.
Question: Is it possible to do something similar to construct a square with area equal to that of a plane figure that has curved sides, instead of straight ones? As a "check" of the construction shown on this page, you can print the page on heavy paper, then cut out the polygon and final square, then weigh them on a sensitive balance. If their areas are the same, and the paper is uniform, and they're cut our perfectly, their weights will be the same.