




"""simple docstring"""





from itertools import count
from math import asin, pi, sqrt

def 		lowerCAmelCase				(			lowerCamelCase		):
						'''simple docstring'''


						return (
						    (1 - 2 * point) * sqrt(point - point**2		) + 2 * point + asin(sqrt(1 - point		)		)
						) / 4

def 		lowerCAmelCase				(			lowerCamelCase		):
						'''simple docstring'''


						snake_case__							:       List[Any]				      =       (circles_number + 1 - sqrt(2 * circles_number		)) / (
						    2 * (circles_number**2 + 1)
						)
						snake_case__							:       Tuple				      =       circles_number * intersection_y

						snake_case__							:       str				      =       intersection_x * intersection_y / 2
						snake_case__							:       Optional[Any]				      =       circle_bottom_arc_integral(
						    1 / 2		) - circle_bottom_arc_integral(lowerCamelCase		)

						return triangle_area + concave_region_area

def 		lowerCAmelCase				(			lowerCamelCase = 1 / 1000		):
						'''simple docstring'''


						snake_case__							:       Tuple				      =       (1 - pi / 4) / 4

						for n in count(1		):
												if concave_triangle_area(lowerCamelCase		) / l_section_area < fraction:
																		return n

						return -1


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








"""simple docstring"""





_lowercase						   =     [0.4_85, 0.4_56, 0.4_06]
_lowercase						   =     [0.2_29, 0.2_24, 0.2_25]
_lowercase						   =     [0.5, 0.5, 0.5]
_lowercase						   =     [0.5, 0.5, 0.5]
_lowercase						   =     [0.48_14_54_66, 0.4_57_82_75, 0.40_82_10_73]
_lowercase						   =     [0.26_86_29_54, 0.26_13_02_58, 0.27_57_77_11]


