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你知道圆的面积计算公式是怎么来的吗

Area of a circle, how to get the formula.

πr^2 gives you the area of a circle
圆的面积是πr²
but where does πr^2 come from ?
但这个公式是怎么来的呢?
first, we’ll draw circle and fill in it’s area
首先 画一个圆并填充颜色
next we will divide it into large equal parts
然后我们把它分成几大等份
and arrange them in a rectangular formation
然后排列成长方形的形状
as you can see it barely resembles a rectangle
能看出 它不太像长方形
so next we will divide the circle into small equal pieces
所以接下来我们要把圆分成几个小等份
and we will arrange them in the same manner
然后用同样的方式排列它们
you can see that it appears more like a rectangle
这时你就可以看到它更像一个长方形了
so if we divide the circle into even more smaller pieces
那么 如果我们将圆分成更多更小的等份
you can see that every time the shape becomes more like a rectangle
可以看到形状每次都更接近一个长方形
so how small must we divide a circle
那么 要得到一个完美的长方形
before we can get a perfect rectangle?
需要把圆分成多小的等份呢?
well, we can keep on dividing the circle into small, smaller,
我们继续将圆等分得越来越小
or the smallest pieces you can make
或者说 将其等分成最小
but the answer is to divide the circle infinitely many times
但答案就是把圆无限次地等分
until we cannot distinguish the lines
直到我们看不清分割线为止
and eventually the circle can now become a perfect rectangle
最终圆就能变成一个完美的长方形了
so the area of the circle
因此 圆的面积
is equal to the area of the rectangle
就等于这个长方形的面积
which is equal to Base times Height
即等于底乘以高
the height of the rectangle
那么长方形的高
As you can see is the same as the radius
正如你所看到的 与圆的半径一样长
so height is equal to radius
所以长方形的高就等于圆的半径
to find the base
至于底
we need to look at the circumference of the circle
我们需要查看圆的周长
and when we compared the base and the circumference
当我们把长方形的底与圆的周长相比
we can see that the base is equal to
可以得出底长就等于
one-half the circumference
圆形周长的一半
remember that the circumference is equal to 2πr
我们都知道 圆的周长等于2πr
when we combine it with one-half
将1/2代入圆周长公式
the two’s cancel out
这两个2就抵消了
and base is equal to πr
于是底就等于πr
so now base times height becomes πr times r
所以现在底乘以高就变成了πr乘以r
combine the r’s together and we have πr^2
将两个r合并在一起就得到了πr²
which is equal to the area of the rectangle
即长方形的面积
which is equal to the area of the circle
也是圆的面积
so πr^2 can be used to calculate
因此 πr²可以用于计算
the area of any size circle that exists
任何大小圆的面积

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视频概述

在本视频中,你会对圆的面积计算有一个更为深入的了解

听录译者

收集自网络

翻译译者

ollivis

审核员

审核员A

视频来源

https://www.youtube.com/watch?v=YokKp3pwVFc

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