def concave_triangle_area(circles_number: int) -> float:

    intersection_y = (circles_number + 1 - sqrt(2 * circles_number)) / (
        2 * (circles_number**2 + 1)
    )
    intersection_x = circles_number * intersection_y

    triangle_area = intersection_x * intersection_y / 2
    concave_region_area = circle_bottom_arc_integral(
        1 / 2
    ) - circle_bottom_arc_integral(intersection_x)

    return triangle_area + concave_region_area