A Talmudic Approach to the Area of a Circle

Abstract

Most elementary approaches to proving that the area of a circle is iirr2 employ various sorts of approximation techniques. For example, the circle is repeatedly bisected by diameters until the resulting sections resemble triangles which when placed together yield the desired result [1, pp. 466-467], or by showing that a circle becomes indistinguishable from a regular polygon as the number of sides increases [2, p. 22]. The primary objection to these methods lies in the fact that a certain degree of mathematical sophistication is required to appreciate the concept of the limiting triangle or limiting regular polygon, and this is especially difficult to convey to students who are not familiar with the limit concept in calculus. An imaginative technique which avoids this pitfall and is readily reproduceable with simple props in a classroom can be found in the commentaries of the Talmud (Tosfos Pesachim 109a, Tosfos Succah 8a, Marsha Babba Bathra 27a), where the need for explaining geometrical formulas to people lacking in mathematical sophistication was paramount. Consider a circle of radius r as being composed of a large number of smaller concentric circles (strings), evenly spaced from each other, starting from the center of the circle (a point) and emanating outwards. If one were then to slice the circle from the top to the origin along a radius and pull each of the strings taut parallel to the bottom tangent of the circle (FIGURE 1) while still maintaining the