英语版数学公式Perimeter周长公式

2023年12月14日发(作者：朋友数)

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Perimeter

Perimeter is the distance around a two-dimensional shape.

Example: the perimeter of this rectangle is 7+3+7+3 = 20

Example: the perimeter of

regular pentagon is 3+3+3+3+3 = 5×3 = 15

The perimeter of a circle is called the circumference:

Circumference = 2π × radius

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Perimeter Formulas

Triangle

Perimeter = a + b + c

Square

Perimeter = 4 × a

a = length of side

Rectangle

Perimeter = 2 × (w + h)

w = width

h = height

Quadrilateral

Perimeter = a + b + c + d

Circle

Circumference = 2πr

r = radius

Sector

Perimeter = r(θ+2)

r = radius

θ = angle in radians

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精选文档 Perimeter of an Ellip

On the Ellip page we looked at the definition and some of the simple

properties of the ellip, but here we look at how to more accurately calculate

its perimeter.

Perimeter

Rather strangely, the perimeter of an ellip is very difficult to calculate!

There are many formulas, here are some interesting ones. (Also e Calculation

Tool below.)

First Measure Your Ellip!

a and b are measured from the center, so they are like "radius" measures.

Approximation 1

This approximation is within about 5% of the true value, so long as a is not more

than 3 times longer than b (in other words, the ellip is not too "squashed"):

精选文档 Approximation 2

The famous Indian mathematician Ramanujan came up with this better

approximation:

Approximation 3

Ramanujan also came up with this one. First we calculate "h":

Then u it here:

Infinite Series 1

This is an exact formula, but it needs an "infinite ries" of calculations to

be exact, so in practice we still only get an approximation.

First we calculate

e (the "eccentricity", not Euler's number "e"):

Then u this "infinite sum" formula:

Which may look complicated, but expands like this:

精选文档 精选文档 The terms continue on infinitely, and unfortunately we must calculate a lot of terms

to get a reasonably clo answer.

Infinite Series 2

But my favorite exact formula (becau it gives a very clo answer after only

a few terms) is as follows:

First we calculate "h":

Then u this "infinite sum" formula:

(Note: the is the Binomial Coefficient with

half-integer factorials ... wow!)

It may look a bit scary, but it expands to this ries of calculations:

The more terms we calculate, the more accurate it becomes (the next term is

4525h/16384, which is getting quite small, and the next is 49h/65536, then

6441h/1048576)

Comparing精选文档 Just for fun, I calculate the perimeter using the three approximation formulas,

and the two exact formulas (but only the first four terms including the "1", so

it is still just an approximation) for the following values of a and b:

Circle

a: 10

b: 10

Approx 1: 62.832

Approx 2: 62.832

Approx 3: 62.832

Series 1: 62.832

Series 2: 62.832

Exact*:

20π

* Exact:

•

10

5

49.673

48.442

48.442

48.876

48.442

10

3

46.385

43.857

43.859

45.174

43.859

10

1

44.65

40.606

40.639

43.204

40.623

Lines

10

0

44.429

39.834

39.984

42.951

39.884

40

When a=b, the ellip is a circle, and the perimeter is 2πa (

in our example).

•

When b=0 (the shape is really two lines back and forth) the perimeter

is 4a (40 in our example).

They all get the perimeter of the circle correct, but only Approx 2 and

3 and Series 2 get clo to the value of 40 for the extreme ca of b=0.

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Ellip Perimeter Calculations Tool

This tool does the calculations from above, but with more terms for the Series.

精选文档 周长

周长是围绕一个二维形状的距离。

例如：此矩形的周长是7 + 3 + 7 + 3 = 20

例如：此常规的周边五边形是3 + 3 + 3 + 3 + 3 = 5×3 = 15

一个的周缘圆圈被称为圆周：

圆周= 2个π ×半径

精选文档 周边公式

三角形

平方

周长= A + B + C

周长= 4×一个

一个=边的长度

矩形

周长= 2×（W + H）

W =宽度

H =高度

四边形

周长= A + B + C + D

圆

周长= 2

π - [R

R =半径

扇区

周长= R（θ+ 2）

R =半径

θ=在角度弧度

椭圆

周长= 很辛苦！

精选文档 椭圆的周长

在椭圆页面我们看到了定义和一些椭圆的简单性质的，但在这里我们就来看看如何更准确地计算出它的周长。

周长

而是奇怪的，椭圆的周长是很难计算！

有许多公式，这里有一些有趣的。（另请参阅计算工具下文）。

先测量你的椭圆！

一和b被测量从中心，因此它们像“半径”的措施。

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逼近1

这种近似是内真值的约5％，所以只要一个长于不超过3倍b（换言之，椭圆是不是太“挤压”）：

逼近2

著名的印度数学家拉马努金想出了这个更好的近似：

逼近3

拉马努金也来到了这一个。首先，我们计算出“H”：

然后在这里使用它：

精选文档 无穷级数1

这是一个 精确的公式，但它需要计算的“无穷级数”是准确的，因此在实践中，我们仍然只得到一个近似。

首先，我们计算Ë （下称“ 偏心 ”，不是 欧拉数“E”）：

然后用这个“无限之和”的公式：

这可能看起来复杂，但扩展是这样的：

该条款继续无限，不幸的是，我们必须计算很多方面的得到一个相当接近的答案。

无穷级数2

但我最喜欢的精确公式（因为它仅提供了一些术语后非常密切的答案）如下：

首先，我们计算出“H”：

然后用这个“无限之和”的公式：

（注：是 二项式系数 与半整数阶乘 ...哇！）精选文档

它可能看起来有点吓人，但它扩展到这一系列的计算：

越术语我们计算，则成为更准确的（下一个项是25 ħ

4 /16384这是越来越相当小的，并且下一个是49 ħ

5/65536然后441 ħ

6 /1048576）

对比

只是为了好玩，我计算出使用三个近似公式周长，两个精确的公式（但只有前四项，包括“1”，所以它仍然只是一个近似值）为下列值一和b：

圈

A： 10

b： 10

约1： 62.832

约2： 62.832

约3： 62.832

系列一： 62.832

系列二： 62.832

精确*：

20个π

*精确：

•

•

10

五

49.673

48.442

48.442

48.876

48.442

10

3

46.385

43.857

43.859

45.174

43.859

10

1

44.65

40.606

40.639

43.204

40.623

行

10

0

44.429

39.834

39.984

42.951

39.884

40

当A = B时，椭圆为圆形，且所述周边是2

π一个（62.832 ...在我们的例子）。

当B = 0（形状实际上是两个线来回）周长为4A（40在我们的例子）。

他们都得到了圆的周长是正确的，但只有约2,3和系列2获得接近40 B = 0的极端情况下的价值。精选文档

椭圆外周计算工具

此工具会计算从以上，但对于更多的系列条款。

（注：可编辑下载，若有不当之处，请指正，谢谢!）

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