query: How can I improve my explanation of why the ratio $\pi=\frac{C}{d}$ holds for all circles?
positive_passages.docid: 155241
positive_passages.text: How do we know the ratio between circumference and diameter is the same for all circles?
negative_passages.docid: 137391
negative_passages.text: How does the unit circle work for trigonometric ratios of non-acute angles?
negative_passages.docid: 75731
negative_passages.text: Significance of $\pi$ in physics
negative_passages.docid: 96805
negative_passages.text: Why is the Absolute Value of $3-π$ equal to $π - 3$
negative_passages.docid: 12841
negative_passages.text: Ratio of sines given, find ratio of cosines?
negative_passages.docid: 11338
negative_passages.text: How to find the circumcenter of a triangle and the length of the corresponding radius of the circle in $\mathbb{R}^3$
negative_passages.docid: 96265
negative_passages.text: Why does $x= \frac{1}{2}(z+\bar{z}) = \frac{1}{2}(z+\frac{r^{2}}{z})$ on the circle?
negative_passages.docid: 105973
negative_passages.text: how do we know the BBP formula for $\pi$ is valid?
negative_passages.docid: 37642
negative_passages.text: Why is $\pi$ irrational if it is represented as $C/D$?
negative_passages.docid: 114734
negative_passages.text: Prove that $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$ where $\mathcal{C}$ is the unit circle
negative_passages.docid: 146530
negative_passages.text: If $z$ lies on the circle $|z-1|=1,$ then the value of $\frac{z-2}{z}$ is
negative_passages.docid: 48086
negative_passages.text: Why is this series of square root of twos equal $\pi$?
negative_passages.docid: 28890
negative_passages.text: Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian?
negative_passages.docid: 69803
negative_passages.text: An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$
negative_passages.docid: 49742
negative_passages.text: Calculating the position of points in a circle
negative_passages.docid: 30429
negative_passages.text: How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?
negative_passages.docid: 151937
negative_passages.text: Physical Significance of $8\pi r$