A Picture is Worth a Thousand Words: Area of Circles
by Corina Murg
We just talked about π. It is only befitting that area of circles follows with an elegant proof from Rabbi Abraham Bar Hiyya Ha-nasi, a Spanish-Jewish scientist and philosopher that lived during the last part of the 11th century and the first part of the 12th century in England. One of the books he published The Book of Mensuration of the Earth and its Division became a textbook of reference in European schools under its Latin translation Liber Embadorum. In this book the reader discovers a surprisingly simple visual proof that the area of a circle is the same as the area of a triangle with height equal to the radius of the circle C and base equal to the circumference of C.
It was not the first time a mathematician had discussed this relationship. Around 225 BC Archimedes had already proved it in his treatise Measurement of the Circle. In this text, Proposition 1 states that every circle is equal to a right-angled triangle, whose radius is equal to one of the sides around the right angle while the circumference is equal to the base of the triangle. Archimedes proved this proposition by the method of exhaustion, showing that the area of the circle can not be greater nor smaller than the area of the triangle, therefore it must be equal to it.
What sets Bar Hiyya’s visual proof apart form Archimedes work is its simplicity and its appeal to a less skilled student of mathematics. Bar Hiyya thought of the region inside a circle C as an area consisting of layers of inner circles. If we cut along a radius and then each inner circle is peeled off and flattened, together they form a triangle of height equal to the radius of the circle C and base equal to the circumference of C.
The figures below depict three different stages of the peeling of the circle.
(They were generated by a Java applet built by Alexander Bogomolny.)