The Shoelace formula (also called Gauss's area formula) is a convenient method to calculate the area of a simple polygon (one that does not intersect itself) when you know the coordinates of its vertices.

If a polygon has $n$ vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$, listed in order (clockwise or counterclockwise), then the area $A$ is: $$ A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right| $$ where $(x_{n+1}, y_{n+1}) = (x_1, y_1)$ - that is, the first vertex is repeated at the end to "close the loop".

The following Python script defines a class that computes the area of a polygon using the Shoelace formula and checks whether the polygon is convex.

class Polygon:
    def __init__(self, vertices):
        self.vertices = vertices

    def get_area(self):
        # Initializations
        area = 0.0
        n = len(self.vertices)
        j = n - 1

        for i in range(0, n):
            area += (self.vertices[j][0] + self.vertices[i][0]) * \
                    (self.vertices[j][1] - self.vertices[i][1])
            j = i   # set j to the previous vertex

        return abs(0.5 * area)

    def is_convex(self):
        # Initializations
        N = len(self.vertices)  # Stores count of edges in polygon
        prev = 0                # Stores direction of cross product of previous traversed edges
        cur = 0                 # Stores direction of cross product of current traversed edges
     
        for i in range(N):
            # Stores three adjacent edges of the polygon
            temp = [self.vertices[i], self.vertices[(i + 1) % N], self.vertices[(i + 2) % N]]
            cur = self.cross_product(temp)
     
            if (cur != 0):
                # If direction of cross product of all adjacent edges are not same
                if (cur * prev < 0):
                    return False
                else:
                    prev = cur
     
        return True

    def cross_product(self, a):
         
        x1 = (a[1][0] - a[0][0])    # Stores abscissas of vector a[1] ---> a[0]
        y1 = (a[1][1] - a[0][1])    # Stores ordinates of vector a[1] ---> a[0]
        x2 = (a[2][0] - a[0][0])    # Stores abscissas of vector a[2] ---> a[0]
        y2 = (a[2][1] - a[0][1])    # Stores ordinates of vector a[2] ---> a[0]
     
        # Return cross product
        return (x1 * y2 - y1 * x2)


if __name__ == "__main__":

    vertices1 = [[1, 6], [3, 1], [7, 2], [2, 4], [8, 5]]
    polygon1 = Polygon(vertices1)
    print(polygon1.get_area())
    print(polygon1.is_convex())

    vertices2 = [[4, 10], [9, 7], [11, 2], [2, 2]]
    polygon2 = Polygon(vertices2)
    print(polygon2.get_area())
    print(polygon2.is_convex())