The calculation of the circumference of a circle was a popular problem in antiquity, and different cultures developed various solutions. In the second century BCE, Archimedes, and later Zu Chongzhi in the fifth century AD, produced remarkably accurate approximations of \(\pi\) . In this chapter, we aim to approximate \(\pi\) using rational numbers, that is, fractions—something the Archimedean approach cannot provide directly. We therefore turn to a completely different method developed in the 17th century: arctangent identities. These equations allow us to construct approximation formulas for \(\pi\) that involve only rational numbers. Good approximations of the arctangent function can be obtained using the so-called Gregory series. Moreover, it turns out to be relatively easy to construct new arctangent identities, leading to new approximations of \(\pi\) . We also carry out systematic searches for particularly “good” approximations. Overall, this classical method based on arctangent identities shows considerable potential and is well worth further study.