Python calculus of area of polygons given vertices

Question

I am trying to calculate the area of a particular polygon.

Let me explain better. I am trying to learn how to deal with polygons in Python. I know there are available packages like Shapely, but I would like to write the code myself using more common packages like numpy, scipy or math.

I am currently trying this exercise: Given some vertices coordinates and one radius measure, calculate the area of the polygon you obtain. The area is formed by: (area of the circle with center in one given point and radius given) - (area of the polygon formed by the given vertices in clockwise).

Let's see some examples:

1. I have a circle and the given vertices form a polygon which does not intersect the circle. The wanted area is just the area of the circle: area = areaCircle (easy task - done).

2. I have a circle and the given vertices form a polygon which does intersect the circle at some point. The radius is like a stick (rigid, its measure doesn't change overtime). I should calculate area = areaCircle - areaPolygon (medium task).

3. I have a circle and the given vertices form a polygon which does intersect the circle at some point. The radius is like a rope (flexible). When the two polygons intersect, the radius varies (so the circle won't be perfect). As you can see, the center of the circle is (5,5) and it has radius ~ 4. When the radius-rope hits the polygon in (4, 7) and (8, 4), it deforms the circle. So, the area I am trying to obtain should be: area = areaDeformedCircle - areaPolygon (hard task).

Here's, so far, my code:

from math import pi

# parser for input
def parser():
    while 1:
        data = list(input().split(' '))
        for number in data:
            if len(number) > 0:
                yield (number)

input_parser = parser()

def get_word():
    global input_parser
    return next(input_parser)

def get_number():
    data = get_word()
    try:
        return int(data)
    except ValueError:
        return float(data)

# input data: 
X,Y,L = map(int,input().split()) # circle data: X_coord_center, Y_coord_center, radius
N = get_number() # number of vertices of the polygon
vertexes = []
for i in range(N): #vertices of the polygon, given clockwise
    xy = (get_number(), get_number())
    vertexes.append(xy)

def circleArea(radius):
    return pi * radius ** 2

# from question https://stackoverflow.com/questions/24467972/calculate-area-of-polygon-given-x-y-coordinates
def polyArea(coords):
    t=0
    for count in range(len(coords)-1):
        y = coords[count+1][1] + coords[count][1]
        x = coords[count+1][0] - coords[count][0]
        z = y * x
        t += z
    return abs(t/2.0)

# area polygon
print("area polygon: ", polyArea(vertexes))

# area circle
print("area circle:  ", circleArea(L))

The thing is, I am only able to do point 1) between those (calculate a circle's radius, no big deal). in fact, given these input examples, I am able to calculate the correct area only when the polygon does not intersect the circle.

example where the circle does not intersect the polygon (answerArea = 3.14159)
3 3 1
4
3 5
6 7
8 5
7 2

example where the circle intersects the polygon (answerArea = 36.71737) - it's the image's data
5 5 4
4
4 7
7 9
9 7
8 4

How should I modify my code (NOT using Shapely) in order to calculate one of the other two points? I don't have a teacher, I'm learning all by myself and online hints didn't help me much.

Thanks in advise to who will help and happy coding to everyone :)