# Pi 是永恆 (一)

$C=2\pi R$

$S=\int \textrm{d}s=\int\sqrt{\textrm{d}x^2+\textrm{d}y^2}$

$S=\int\textrm{d}s=\int\sqrt{1+\left(\frac{\textrm{d}y}{\textrm{d}x}\right)^2}\textrm{d}x$

$\frac{\textrm{d}y}{\textrm{d}x}\times\frac{y}{x}=-1$

$\frac{\textrm{d}y}{\textrm{d}x}=-\frac{x}{y}$

$S=\int\sqrt{1+\left(\frac{-x}{y}\right)^2}\textrm{d}x$

$S=\int\frac{R}{y}\textrm{d}x$

$S=\int 1 / \sin\theta\textrm{d}x$

$\textrm{d} x = - R \sin\theta \textrm{d}\theta$

$S=\int -R\textrm{d}\theta$

$S/2R=\int -1/2\textrm{d}\theta$

$(C/2)/2R=\int^{\pi}_{0} -1/2\textrm{d}\theta = -\pi/2$

$C=2\pi R$

$\pi$ 是永恆。下回，我們來看看它如何能夠 loop 到下個世紀，仲未埋尾。

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Categorized as 科普

## By David Yu

David Yu is the father of Simba and an Astrophysicist. He obtained his PhD in Astrophysics from the Max Planck Institute of Extraterrestrial Physics, Germany. He was formerly a postdoc at the KTH Royal Institute of Technology, Sweden, and is currently a lecturer at the University of Hong Kong. He served as the host of the RTHK TV31 programme Science Night, the Depute Editor-in-Chief of the Taiwanese Science Magazine Physics Bimonthly, and the Science Consultant of Stand News. The books he authored include Dr David’s 21 Mysteries of The Universe (2021), Space – A Down-to-Earth Astrophysicist (2021), Time – A Spacetime Journey of Astronomy and Physics (2019), and Secrets from the Stars (2017).