  Pendulums have long fascinated humans ever since Galileo theorized that they
are isochronic with regards to their swing. While this simplification is useful
in the case of small-angle pendulums due to the accuracy of the small-angle
approximation, it breaks down for large-angle pendulums and can cause larger
problems with the computational modelling of simple pendulums. This paper will
examine the differences between the periods of small-angle and large-angle
pendulums, offering derivations of the period in both models from the basic
laws of nature. This paper also provides a common way of deriving elliptic
integrals from physical phenomena, and the period of pendulums has been one of
the major building blocks in this new, developing field. Lastly, this paper
makes a number of suggestions for extensions into the study of simple pendulums
that can be performed. While this paper is not intended as a rigorous
mathematical proof, it is designed to illuminate the derivation of the exact
periods of simple pendulums and carefully walks through the mathematics
involved.
