


from math import pi, sqrt, tan
def 							UpperCamelCase_(					_A :str	)->		List[str]:
			if side_length < 0:
						raise ValueError("surface_area_cube() only accepts non-negative values"	)
			return 6 * side_length**2
def 							UpperCamelCase_(					_A :List[Any]					,  _A :List[str]					,  _A :List[str]	)->		str:
			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 							UpperCamelCase_(					_A :Optional[int]	)->		Union[str, Any]:
			if radius < 0:
						raise ValueError("surface_area_sphere() only accepts non-negative values"	)
			return 4 * pi * radius**2
def 							UpperCamelCase_(					_A :Any	)->		Dict:
			if radius < 0:
						raise ValueError("surface_area_hemisphere() only accepts non-negative values"	)
			return 3 * pi * radius**2
def 							UpperCamelCase_(					_A :Dict					,  _A :Optional[Any]	)->		Optional[Any]:
			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 							UpperCamelCase_(					_A :str					,  _A :List[str]					,  _A :str	)->		List[Any]:
			if radius_a < 0 or radius_a < 0 or height < 0:
						raise ValueError(
						    "surface_area_conical_frustum() only accepts non-negative values"	)
			UpperCamelCase__       =							(height**2 + (radius_a - radius_a) ** 2) ** 0.5
			return pi * ((slant_height * (radius_a + radius_a)) + radius_a**2 + radius_a**2)
def 							UpperCamelCase_(					_A :List[str]					,  _A :Tuple	)->		Tuple:
			if radius < 0 or height < 0:
						raise ValueError("surface_area_cylinder() only accepts non-negative values"	)
			return 2 * pi * radius * (height + radius)
def 							UpperCamelCase_(					_A :Any					,  _A :Optional[Any]	)->		Optional[int]:
			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(_A					,  2	) * torus_radius * tube_radius
def 							UpperCamelCase_(					_A :Any					,  _A :Tuple	)->		Optional[Any]:
			if length < 0 or width < 0:
						raise ValueError("area_rectangle() only accepts non-negative values"	)
			return length * width
def 							UpperCamelCase_(					_A :List[Any]	)->		Union[str, Any]:
			if side_length < 0:
						raise ValueError("area_square() only accepts non-negative values"	)
			return side_length**2
def 							UpperCamelCase_(					_A :str					,  _A :int	)->		List[str]:
			if base < 0 or height < 0:
						raise ValueError("area_triangle() only accepts non-negative values"	)
			return (base * height) / 2
def 							UpperCamelCase_(					_A :int					,  _A :Union[str, Any]					,  _A :Tuple	)->		List[Any]:
			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"	)
			UpperCamelCase__       =							(sidea + sidea + sidea) / 2
			UpperCamelCase__       =							sqrt(
			    semi_perimeter
			    * (semi_perimeter - sidea)
			    * (semi_perimeter - sidea)
			    * (semi_perimeter - sidea)	)
			return area
def 							UpperCamelCase_(					_A :List[Any]					,  _A :List[Any]	)->		Tuple:
			if base < 0 or height < 0:
						raise ValueError("area_parallelogram() only accepts non-negative values"	)
			return base * height
def 							UpperCamelCase_(					_A :List[Any]					,  _A :Optional[Any]					,  _A :Dict	)->		List[Any]:
			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 							UpperCamelCase_(					_A :Any	)->		List[Any]:
			if radius < 0:
						raise ValueError("area_circle() only accepts non-negative values"	)
			return pi * radius**2
def 							UpperCamelCase_(					_A :List[Any]					,  _A :int	)->		Optional[int]:
			if radius_x < 0 or radius_y < 0:
						raise ValueError("area_ellipse() only accepts non-negative values"	)
			return pi * radius_x * radius_y
def 							UpperCamelCase_(					_A :int					,  _A :Optional[int]	)->		Union[str, Any]:
			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 							UpperCamelCase_(					_A :Optional[int]					,  _A :Tuple	)->		Union[str, Any]:
			if not isinstance(_A					,  _A	) 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(1_0, 2_0) = }''')
						print(f'''Square: {area_square(1_0) = }''')
						print(f'''Triangle: {area_triangle(1_0, 1_0) = }''')
						print(f'''Triangle: {area_triangle_three_sides(5, 1_2, 1_3) = }''')
						print(f'''Parallelogram: {area_parallelogram(1_0, 2_0) = }''')
						print(f'''Rhombus: {area_rhombus(1_0, 2_0) = }''')
						print(f'''Trapezium: {area_trapezium(1_0, 2_0, 3_0) = }''')
						print(f'''Circle: {area_circle(2_0) = }''')
						print(f'''Ellipse: {area_ellipse(1_0, 2_0) = }''')
						print('\nSurface Areas of various geometric shapes: \n')
						print(f'''Cube: {surface_area_cube(2_0) = }''')
						print(f'''Cuboid: {surface_area_cuboid(1_0, 2_0, 3_0) = }''')
						print(f'''Sphere: {surface_area_sphere(2_0) = }''')
						print(f'''Hemisphere: {surface_area_hemisphere(2_0) = }''')
						print(f'''Cone: {surface_area_cone(1_0, 2_0) = }''')
						print(f'''Conical Frustum: {surface_area_conical_frustum(1_0, 2_0, 3_0) = }''')
						print(f'''Cylinder: {surface_area_cylinder(1_0, 2_0) = }''')
						print(f'''Torus: {surface_area_torus(2_0, 1_0) = }''')
						print(f'''Equilateral Triangle: {area_reg_polygon(3, 1_0) = }''')
						print(f'''Square: {area_reg_polygon(4, 1_0) = }''')
						print(f'''Reqular Pentagon: {area_reg_polygon(5, 1_0) = }''')



from math import ceil, sqrt






def _lowerCamelCase  (		snake_case = 1_000_000  ):
							_lowerCAmelCase							=							0

							for outer_width in range(3     , (limit // 4) + 2  ):
														if outer_width**2 > limit:
																					_lowerCAmelCase							=							max(ceil(sqrt(outer_width**2 - limit  )  )     , 1  )
														else:
																					_lowerCAmelCase							=							1
														if (outer_width - hole_width_lower_bound) % 2:
																					hole_width_lower_bound += 1

														answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

							return answer


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






