pi.py

from math import sqrt

# 2 constants: the square root of 2, and the raduis of the circle we're
# drawing.
ROOT_2 = sqrt(2)
RADIUS = float(1000000000)

# 2 variables: current x position, and the area of the circle being
# calculated.
x = RADIUS
circle_area = float()

# Loop through y values of 1 through the y value where a 45 degree line is
# drawn from the center. Saving time be drawing only 1/8th of a circle, the
# smallest symetrical section of a crcle in a square.
for y in range(1, int(ROOT_2  * 0.5 * RADIUS) + 1):
    # While the hypotenuse is greater than the radius, deincriment x by 1.
    while sqrt(x ** 2 + y ** 2) > RADIUS:
        x -= 1
    # Add the area of the section of the current line (from the 45 degree line
    # to the edge of the circle) times 8 to the circle's area value.
    circle_area += (x - y) * 8

# using a = πr^2 reverse it -> π = a/r^2 and print out the result.
print(format(circle_area / (RADIUS ** 2), '.100g'))