





from __future__ import annotations

from math import ceil, floor, sqrt





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE = 2_000_000				)							-> int:




    """simple docstring"""




    _A     =							[0]
    _A     =							42

    for idx in range(1							,     ceil(sqrt(target * 2				) * 1.1				)				):
        triangle_numbers.append(triangle_numbers[-1] + idx				)

    # we want this to be as close as possible to target
    _A     =							0
    # the area corresponding to the grid that gives the product closest to target
    _A     =							0
    # an estimate of b, using the quadratic formula
    _A     =							42
    # the largest integer less than b_estimate
    _A     =							42
    # the largest integer less than b_estimate
    _A     =							42
    # the triangle number corresponding to b_floor
    _A     =							42
    # the triangle number corresponding to b_ceil
    _A     =							42

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]							,     1				):
        _A     =							(-1 + sqrt(1 + 8 * target / triangle_a				)) / 2
        _A     =							floor(_SCREAMING_SNAKE_CASE				)
        _A     =							ceil(_SCREAMING_SNAKE_CASE				)
        _A     =							triangle_numbers[b_floor]
        _A     =							triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a				) < abs(
            target - best_product				):
            _A     =							triangle_b_first_guess * triangle_a
            _A     =							idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a				) < abs(
            target - best_product				):
            _A     =							triangle_b_second_guess * triangle_a
            _A     =							idx_a * b_ceil

    return area


if __name__ == "__main__":
 print(f"{solution() = }")












from math import pi, sqrt, tan





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if side_length < 0:
        raise ValueError('surface_area_cube() only accepts non-negative values'				)
    return 6 * side_length**2





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if length < 0 or breadth < 0 or height < 0:
        raise ValueError('surface_area_cuboid() only accepts non-negative values'				)
    return 2 * ((length * breadth) + (breadth * height) + (length * height))





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius < 0:
        raise ValueError('surface_area_sphere() only accepts non-negative values'				)
    return 4 * pi * radius**2





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius < 0:
        raise ValueError('surface_area_hemisphere() only accepts non-negative values'				)
    return 3 * pi * radius**2





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius < 0 or height < 0:
        raise ValueError('surface_area_cone() only accepts non-negative values'				)
    return pi * radius * (radius + (height**2 + radius**2) ** 0.5)





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius_a < 0 or radius_a < 0 or height < 0:
        raise ValueError(
            'surface_area_conical_frustum() only accepts non-negative values'				)
    _A     =							(height**2 + (radius_a - radius_a) ** 2) ** 0.5
    return pi * ((slant_height * (radius_a + radius_a)) + radius_a**2 + radius_a**2)





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius < 0 or height < 0:
        raise ValueError('surface_area_cylinder() only accepts non-negative values'				)
    return 2 * pi * radius * (height + radius)





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if torus_radius < 0 or tube_radius < 0:
        raise ValueError('surface_area_torus() only accepts non-negative values'				)
    if torus_radius < tube_radius:
        raise ValueError(
            'surface_area_torus() does not support spindle or self intersecting tori'				)
    return 4 * pow(_SCREAMING_SNAKE_CASE							,     2				) * torus_radius * tube_radius





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if length < 0 or width < 0:
        raise ValueError('area_rectangle() only accepts non-negative values'				)
    return length * width





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if side_length < 0:
        raise ValueError('area_square() only accepts non-negative values'				)
    return side_length**2





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if base < 0 or height < 0:
        raise ValueError('area_triangle() only accepts non-negative values'				)
    return (base * height) / 2





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if sidea < 0 or sidea < 0 or sidea < 0:
        raise ValueError('area_triangle_three_sides() only accepts non-negative values'				)
    elif sidea + sidea < sidea or sidea + sidea < sidea or sidea + sidea < sidea:
        raise ValueError('Given three sides do not form a triangle'				)
    _A     =							(sidea + sidea + sidea) / 2
    _A     =							sqrt(
        semi_perimeter
        * (semi_perimeter - sidea)
        * (semi_perimeter - sidea)
        * (semi_perimeter - sidea)				)
    return area





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if base < 0 or height < 0:
        raise ValueError('area_parallelogram() only accepts non-negative values'				)
    return base * height





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if basea < 0 or basea < 0 or height < 0:
        raise ValueError('area_trapezium() only accepts non-negative values'				)
    return 1 / 2 * (basea + basea) * height





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius < 0:
        raise ValueError('area_circle() only accepts non-negative values'				)
    return pi * radius**2





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if radius_x < 0 or radius_y < 0:
        raise ValueError('area_ellipse() only accepts non-negative values'				)
    return pi * radius_x * radius_y





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if diagonal_a < 0 or diagonal_a < 0:
        raise ValueError('area_rhombus() only accepts non-negative values'				)
    return 1 / 2 * diagonal_a * diagonal_a





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> float:




    """simple docstring"""




    if not isinstance(_SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				) or sides < 3:
        raise ValueError(
            'area_reg_polygon() only accepts integers greater than or \
equal to three as number of sides'				)
    elif length < 0:
        raise ValueError(
            'area_reg_polygon() only accepts non-negative values as \
length of a side'				)
    return (sides * length**2) / (4 * tan(pi / sides				))
    return (sides * length**2) / (4 * tan(pi / sides				))


if __name__ == "__main__":
 import doctest

 doctest.testmod(verbose=True)  # verbose so we can see methods missing tests

 print("[DEMO] Areas of various geometric shapes: \n")
 print(f"Rectangle: {area_rectangle(10, 20) = }")
 print(f"Square: {area_square(10) = }")
 print(f"Triangle: {area_triangle(10, 10) = }")
 print(f"Triangle: {area_triangle_three_sides(5, 12, 13) = }")
 print(f"Parallelogram: {area_parallelogram(10, 20) = }")
 print(f"Rhombus: {area_rhombus(10, 20) = }")
 print(f"Trapezium: {area_trapezium(10, 20, 30) = }")
 print(f"Circle: {area_circle(20) = }")
 print(f"Ellipse: {area_ellipse(10, 20) = }")
 print("\nSurface Areas of various geometric shapes: \n")
 print(f"Cube: {surface_area_cube(20) = }")
 print(f"Cuboid: {surface_area_cuboid(10, 20, 30) = }")
 print(f"Sphere: {surface_area_sphere(20) = }")
 print(f"Hemisphere: {surface_area_hemisphere(20) = }")
 print(f"Cone: {surface_area_cone(10, 20) = }")
 print(f"Conical Frustum: {surface_area_conical_frustum(10, 20, 30) = }")
 print(f"Cylinder: {surface_area_cylinder(10, 20) = }")
 print(f"Torus: {surface_area_torus(20, 10) = }")
 print(f"Equilateral Triangle: {area_reg_polygon(3, 10) = }")
 print(f"Square: {area_reg_polygon(4, 10) = }")
 print(f"Reqular Pentagon: {area_reg_polygon(5, 10) = }")





