


from math import pi, sqrt, tan






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






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






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






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






def 							__lowercase       (			__lowerCAmelCase							:    float ,   __lowerCAmelCase							:    float       ):
							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(__lowerCAmelCase ,   2       ) * torus_radius * tube_radius






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






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






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






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






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






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






def 							__lowercase       (			__lowerCAmelCase							:    float ,   __lowerCAmelCase							:    float       ):
							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 							__lowercase       (			__lowerCAmelCase							:    int ,   __lowerCAmelCase							:    float       ):
							if not isinstance(__lowerCAmelCase ,   __lowerCAmelCase       ) 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) = }""")








from collections import defaultdict
from math import ceil, sqrt






def 							__lowercase       (			__lowerCAmelCase							:    int = 1_0_0_0_0_0_0 ,   __lowerCAmelCase							:    int = 1_0       ):
							a__					=					defaultdict(__lowerCAmelCase       )

							for outer_width in range(3 ,   (t_limit // 4) + 2       ):
														if outer_width * outer_width > t_limit:
																					a__					=					max(
																					    ceil(sqrt(outer_width * outer_width - t_limit       )       ) ,   1       )
														else:
																					a__					=					1

														hole_width_lower_bound += (outer_width - hole_width_lower_bound) % 2

														for hole_width in range(__lowerCAmelCase ,   outer_width - 1 ,   2       ):
																					count[outer_width * outer_width - hole_width * hole_width] += 1

							return sum(1 for n in count.values() if 1 <= n <= 1_0       )


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




