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可以说圆是宇宙中最基本的形状

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行星的轨道是圆形的

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轮子也是圆形的

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分子层级也是圆的

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圆是无处不在的

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因此我们有必要了解

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圆的一些特性

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人类第一次发现圆形

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可能就是抬头看见的月亮

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人们会问

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圆有什么特性呢？

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第一个特性就是

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圆上所有的点

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到圆心距离相等

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圆上所有点

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到圆心距离相等

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那么有人会问

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这个距离叫什么

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就是到中心都相等的距离

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就是它

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我们称它为圆的半径

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就是圆心到圆边缘的距离

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如果半径是3厘米

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那么这个半径就是3厘米

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这条半径也是3厘米

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这个永远不会变

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圆的定义就是 圆周上的所有点

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到中心的距离都相等

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这个距离就是半径

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那么接下来还有什么有趣的特性

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人们会问圆有多胖？

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也就是圆上两点最远的距离？

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或者从圆上最远的两点把圆切开

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那个距离是多少？

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但它也不一定必须在这

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我可以轻而易举地这样切

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但我不会这样切

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因为这样不是最长的距离

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找到最远的两点切开有很多选择

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我们刚讲了半径

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而最长的距离是从边缘上一点出发经过圆心直到圆上另一点

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所以实际上就是两条半径

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这是一条半径

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这边也是一条半径

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我们称圆上最远两点间的距离

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为直径

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因此这就是圆的直径

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直径与半径的关系很简单

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直径等于半径的二倍

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接下来 大家想知道的可能是

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围绕圆一周有多长？

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因此你用卷尺来量

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就像这样来量

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就像这样来量

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这个距离有多长？

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我们称这个距离为圆的周长

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现在 我们知道了半径与直径的关系

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那么周长与直径有什么关系呢？

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如果不习惯用直径

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搞清半径与周长的关系很简单

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很久很久以前

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人们用卷尺

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不断测量周长与半径

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假设当时的卷尺还没有那么精确

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假设他们测量的这个圆的周长

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差不多是3

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在测量圆的半径

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或者是直径

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直径差不多是1

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他们把数据记下

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他们对这个比例很担心

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让我写下这个

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即周长与直径的比例

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假设他们测量的就是这个圆

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测量的是这个圆

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当时卷尺不是很精确

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他们环绕圆测量

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环绕一周的距离

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大约是3米

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测量直径

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大约是1米

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这很有意思

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那么周长与直径的比例就是3

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因此 有可能所有圆的周长都是直径的三倍

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但只是对这个圆成立

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那么其他的圆呢

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就像这样 我画小一点

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他们测量这个圆

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直径是6厘米

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当时的卷尺不是很精确

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测量显示

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直径大约是2厘米

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再次显示

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周长与半径的比是3

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哇 这个圆的很有趣的特性

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那么有可能 任何圆的周长

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与直径的比例都是一个固定的数值

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因此他们决定进一步研究

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测量也更加精确了

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当测量更加精确后

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确信直径绝对是1

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直径绝对是1

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但是测量周长发现

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周长实际是3.1

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这个圆也是这样

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他们发现周长与直径的比更接近3.1

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随着度量日益精确

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慢慢地就得出这个数字

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测量不断精确

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得出的数字是

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发现这是个无限不循环小数

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这个神奇的数字不断出现

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这个数字对于宇宙来说如此重要

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因为圆在宇宙中很重要

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这个比对于任何圆都成立

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圆周长与直径的比就是

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这个神奇的数字 人们还给它起了个名字

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称之为pi

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或者是拉丁希腊文的π

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这就代表了

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宇宙中这个神奇的数字

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最初这个字母只是代表周长与直径的比

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但随着对数学的了解

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你会发现它还出现在其他地方

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它就是宇宙中最基本的数字之一

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让人们觉得宇宙必有规则所在

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但是 我们数学中怎么用π呢？

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我们知道 周长与直径的比

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但我说比的时候

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就是字面意思 周长除以直径

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就得到π

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π就是指这个数字

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我可以写下3.14159一直无限不循环下去

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但写下去只是浪费空间罢了

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而且也让计算更麻烦了

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因此人们只在这儿写上希腊字母π

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那么 这怎么产生联系呢?

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两边可以都乘以直径

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那么周长就是

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π乘以直径

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又因为直径是半径的二倍

142
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因此周长也是

143
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π乘以2r

144
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更常见的形式则是

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2πr

146
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我们可以用这个公式解决一些问题

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假设有一个圆

148
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我们已知半径

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半径是3

150
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半径是3 我写下来 半径是3

151
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加个单位 就假设是3米

152
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求圆的周长

153
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周长等于2π乘以半径

154
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因此就是 2π乘以半径

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就是乘以3米

156
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也就是6πm

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6πm

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我可以把它算出来

159
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记住π只是一个数字

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π是3.14159无限不循环

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如果6乘以这个数

162
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就过就是18点几几

163
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如果有计算器 你可以算出来

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但是为了简单 人们一般就直接写π

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我也不知道6乘以3.14159具体等多少

166
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结果大概是18 19

167
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差不多就是18点几几

168
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因为我没有计算器

169
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就只写6π

170
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不写具体数字了

171
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实际上 这肯定不会超过19

172
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现在再看一个问题

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圆的直径是多少？

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如果半径是3 那么直径就是半径的二倍

175
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也就是3乘以2 或 3加3

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就是6米

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因此周长是6πm 直径是6米

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半径是3米

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再换一个方式看

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假设另外有一个圆

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这有另外一个圆

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已知周长是10m

183
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10πm就是周长

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如果用卷尺量圆周长

185
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求圆的直径

186
00:09:19,430 --> 00:09:21,290
我们知道直径乘以π

187
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π乘以直径等于周长

188
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也就是10米

189
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因此两边都除以π

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就解决问题了

191
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直径就是10m/π

192
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π只是一个数字

193
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如果你有计算器

194
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你可以用10除以3.14159

195
00:09:45,720 --> 00:09:47,880
你会得到3点多米

196
00:09:48,440 --> 00:09:49,400
我心算不出来

197
00:09:49,860 --> 00:09:50,760
这是一个数字

198
00:09:51,100 --> 00:09:53,160
但为了简单 我就只写π

199
00:09:53,530 --> 00:09:55,620
那么半径呢？

200
00:09:56,390 --> 00:09:58,550
半径是直径的一半

201
00:10:00,550 --> 00:10:03,020
这个距离是10/π

202
00:10:03,880 --> 00:10:06,430
如果求半径

203
00:10:07,150 --> 00:10:08,220
乘以1/2就行了

204
00:10:08,500 --> 00:10:11,510
就是1/2乘以10除以π

205
00:10:12,900 --> 00:10:14,190
就是1/2乘以10

206
00:10:14,540 --> 00:10:18,460
约去2

207
00:10:19,430 --> 00:10:21,000
就是5 就是5/π

208
00:10:22,120 --> 00:10:24,110
半径就是5/π

209
00:10:25,050 --> 00:10:26,480
这没什么神奇的

210
00:10:27,070 --> 00:10:29,760
我觉得最迷惑人的是

211
00:10:30,220 --> 00:10:32,070
π是一个数字

212
00:10:32,780 --> 00:10:37,400
π是3.14159无限不循环

213
00:10:37,770 --> 00:10:42,070
有几千本书都研究过π

214
00:10:42,920 --> 00:10:45,350
也许没有几千本

215
00:10:45,610 --> 00:10:46,980
我有点夸张了

216
00:10:47,220 --> 00:10:48,820
但是你可以写关于π的书

217
00:10:49,410 --> 00:10:50,010
它只是一个数字

218
00:10:50,300 --> 00:10:51,180
一个独特的数字

219
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如果你想按照

220
00:10:53,840 --> 00:10:54,770
平常写数字的方法

221
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你可以把这个约掉

222
00:10:56,520 --> 00:10:59,330
但是人们往往

223
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就直接写π

224
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不管了 我也要把π写在这儿

225
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下次 我将讲圆的面积