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General Science / Question related with pi number
« on: 05/01/2021 11:52:49 »
Can you help me please with question assuming attached file. We assume that the radius is equal to 1. It is seen from the drawing that the sum of 2 others described will be selected on either side of the inscribed n-gon. But according to the well-known theorem, any side of a triangle is less than the sum of 2 other sides, which obviously implies a constant less than the inscribed n-gon in comparison with the described one. But we also see from the drawing:
2 * x [3] = (1-x [3]) * sqrt (2), whence x [3] = sqrt (2) / (2 + sqrt (2)), also x [1] = 1-sqrt (1- (2-sqrt (2)) / 4) and x [2] = sqrt (1 + 1 / (3 + 2 * sqrt (2))) - 1. Since x [1]> x [2] and each side of the inscribed n-gon has a side equal to the sum of the two sides of the triangle containing x [1] and x [2] for the described n-gon, we conclude that the limit of the described n -gon, although like the inscribed n-gon has a circle, the circumscribed n-gon tends to 2 * pi faster than the inscribed n-gon. As far as I know, no one paid attention to this fact earlier, trying to approve the concept of calculating the circumference of a circle in the wrong way by finding the supposedly zero limit of the difference between the variable perimeters of the described and inscribed polygons, assuming this while increasing the number of their sides. in this regard, I am sure that the limit of the difference between the circumscribed and inscribed polygons is not equal to 0. Am I right? Thanks a lot!
2 * x [3] = (1-x [3]) * sqrt (2), whence x [3] = sqrt (2) / (2 + sqrt (2)), also x [1] = 1-sqrt (1- (2-sqrt (2)) / 4) and x [2] = sqrt (1 + 1 / (3 + 2 * sqrt (2))) - 1. Since x [1]> x [2] and each side of the inscribed n-gon has a side equal to the sum of the two sides of the triangle containing x [1] and x [2] for the described n-gon, we conclude that the limit of the described n -gon, although like the inscribed n-gon has a circle, the circumscribed n-gon tends to 2 * pi faster than the inscribed n-gon. As far as I know, no one paid attention to this fact earlier, trying to approve the concept of calculating the circumference of a circle in the wrong way by finding the supposedly zero limit of the difference between the variable perimeters of the described and inscribed polygons, assuming this while increasing the number of their sides. in this regard, I am sure that the limit of the difference between the circumscribed and inscribed polygons is not equal to 0. Am I right? Thanks a lot!