Calculate Pi to a defined number of decimal places using Archimedes method.

pi.py

from decimal import Decimal, getcontext

def pi_archimedes(n):
    """
    Calculate Pi using Archimedes method with n iterations to estimate Pi.
    This method approximates Pi by calculating the perimeter of a polygon inscribed
    within a unit circle.

    Polygon edge lengths are computed using the Pythagorean theorem and the geometry of the polygons. 
    The number of sides of the polygon is also doubled in each iteration, as each side of the 
    polygon is bisected to form a new polygon with twice as many sides.

    The formula is:
        sides = 2 * 2^n
        length^2 = 2 - 2 * sqrt(1 - length^2 / 4))

    After n iterations, the function returns the approximate value of Pi using the formula:
        perimeter = sides * sqrt(length^2)
    """
    polygon_edge_length_sq = Decimal(2)
    polygon_sides = 2
    # Start with a line, then a square, then a octagon, etc.
    for _ in range(n):
        polygon_edge_length_sq = 2 - 2 * (1 - polygon_edge_length_sq / 4).sqrt()
        polygon_sides = polygon_sides * 2
    return polygon_sides * polygon_edge_length_sq.sqrt()

# Set the number of decimal places to calculate
PLACES = 100

# Calculate Pi with increasing iterations until the result converges at
# the desired number of decimal places
old_result = None
for n in range(10*PLACES):
    getcontext().prec = 2 * PLACES  # Do calculations with double precision
    result = pi_archimedes(n)
    getcontext().prec = PLACES      # Print the result with single precision
    result = +result                # Rounding
    print("%3d: %s" % (n, result))
    if result == old_result:        # Did it converge?
        break
    old_result = result