Measuring the Accuracy of an Ancient Area Formula

Abstract

In the ancient world, geometers were concerned primarily with mensuration (the practice of accurate measurement), with the most obvious applications being in construction and surveying. One well-known formula from this time (appearing most famously in the Temple of Horus in Egypt, c. 237-57 BC) purports to give the area of a general quadrilateral by averaging the lengths of opposite pairs of sides and then multiplying the averages. While this formula is erroneous, it produces highly accurate results when the quadrilateral is nearly rectangular. We examine the relative accuracy of this area formula, including: (i) different methods of finding the exact area, (ii) how to find the interior angle that minimizes the error of the formula, and (iii) how significantly the error varies as the interior angle varies from the ideal. In the end, these observations lead to an improved version of the formula that relies only on the side-lengths of a quadrilateral.