Archimedes approximated the value of Pi by starting with the fact that a regular hexagon inscribed in a unit circle has a perimeter of 6. He then found a method for finding the perimeter of a polygon with twice as many sides. Applying his method repeatedly, he found the perimeter of a 12, 24, 48, and 96 sided polygon. Using the perimeter as an approximation for the circumference of a circle he was able to derive an approximation for Pi equivalent to 3.14. This video uses a somewhat simpler method of doing the same thing and carries it out to polygons with millions of sides. All that is needed to understand the calculation is knowledge of the Pythagorean Theorem.

For a followup (showing both the inscribed and circumscribed polygons) see    • Finding Pi by Archimedes' Method (Fol...  
For a good explanation of Archimedes actual steps used to carry out his calculation see
https://itech.fgcu.edu/faculty/clinds...
See my website: https://mathwithoutborders.com