Archimedes' Method for Computing Areas and Volumes - Introduction

Gabriela R. Sanchis

We owe many familiar area and volume formulas to Archimedes. For example, he proved that the area \(A\) of a circle equals half its circumference \(C\) multiplied by the radius \(r,\) i.e. \(A=\frac{1}{2}Cr.\)

This formula is often "explained" by the following picture, which shows that if you cut up the circle into an even number of equal sectors, the sectors can be reassembled into a figure that looks approximately like a parallelogram whose base is half the circumference of the circle and whose height is the radius. Click on the image below to see an animation.

Archimedes derived many formulas that are familiar to us today for computing relationships among volumes of spheres, cylinders, and paraboloids. How was he able to discover these formulas? About one hundred years ago, an old Greek manuscript containing works by Archimedes was found which explained his Method, based on the Law of the Lever. The exciting story of this manuscript can be found here.

Editor's note: This article was originally published in Convergence in 2005. We are pleased to "reprint" it in June 2016 with GeoGebra applets hosted by GeoGebraTube. In addition to viewing these applets in the webpages of this article, you may view and download all of the applets that appear in the article directly from GeoGebraTube at