Pi ($\pi$)

$\pi$ is the celebrity constant of geometry—showing up from circles to Fourier series.

Definition

Circle ratio

$\pi = \dfrac{C}{d}$, where $C$ is a circle’s circumference and $d$ is its diameter. The ratio is constant for all circles.

A Lightning History

• Ancient Babylonia: $\pi \approx 3.125$; ancient Egypt: $\pi \approx 3.1605$.
• Archimedes (3rd c. BCE): $3.1408 < \pi < 3.1429$ via polygon approximation.
• Zu Chongzhi (5th c.): $\pi \approx 3.1415926$, centuries ahead of his time.
• Euler popularized the Greek letter $\pi$ in the 18th century.
• Modern computing has pushed known digits past the trillion mark.

Useful Facts

• $\pi$ is irrational and transcendental.
• Key angle values: $\sin\left(\dfrac{\pi}{6}\right) = \dfrac{1}{2}$, $\cos\left(\dfrac{\pi}{3}\right) = \dfrac{1}{2}$.
• Circumference: $C = 2\pi r$; Area: $A = \pi r^2$.
• Period of $\sin x$ and $\cos x$ is $2\pi$.

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练习题

练习 1

Convert $180^\circ$ to radians and $2\pi$ radians to degrees.

参考答案

$180^\circ = \pi$ radians; $2\pi$ radians $= 360^\circ$.

练习 2

Find the area of a circle with radius $3$.

参考答案

$A = \pi r^2 = \pi \times 3^2 = 9\pi$.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\pi$	希腊字母	Pi（派）	圆周率
$C$	数学符号	circumference	圆的周长
$d$	数学符号	diameter	圆的直径
$r$	数学符号	radius	圆的半径

中英对照

中文术语	英文术语	音标	说明
圆周率	pi	/paɪ/	圆的周长与直径之比
直径	diameter	/daɪˈæmɪtər/	穿过圆心的线段长度
半径	radius	/ˈreɪdiəs/	圆心到圆周的距离
周长	circumference	/sərˈkʌmfərəns/	围绕圆的长度
面积	area	/ˈeəriə/	圆的面积 $A=\pi r^2$

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