Approximate PI Geometrically
To calculate π you can use a n-sided polygon to approximate the circumference of a circle. Then apply the definition of π: the circumference of the circle divided by its diameter.
The size of the circle does not matter as the circumference and diameter will be scaled by the same amount. The result of this approach is visualized below.
However, finding the perimeter of a n-sided polygon without using π itself is difficult. Instead, we can make the problem simpler by restricting the number of sides to be a number of base two: 4, 8, 16, etc. Then we can use the formula for a n to the two sided polygon to calculate the perimeter:
Using this formula we can calculate π for the first five polygons.