英语版数学公式Perimeter周长公式

PerimeterPerimeter is the distance around a two-dimensional shape.Example: the perimeter of this rectangle is7+3+7+3=20Example: the perimeter of this regularpentagonis3+3+3+3+3=53 = 15The perimeter of acircleis called the circumference:Circumference = 2 radiusPerimeter FormulasTrianglePerimeter = a + b + cSquarePerimeter = 4 aa = length of sideRectanglePerimeter = 2 (w + h)w = widthh = heightQuadrilateralPerimeter = a + b + c + dCircleCircumference = 2rr = radiusSectorPerimeter = r(+2)r = radius = angle inradiansPerimeter of an EllipseOn theEllipsepage we looked at the definition and some of the simple properties of the ellipse, but here we look at how to more accurately calculate its perimeter.PerimeterRather strangely, the perimeter of an ellipse isvery difficult to calculate!There are many formulas, here are some interesting ones. (Also seeCalculation Toolbelow.)First Measure Your Ellipse!aandbare measuredfrom the center, so they are like radius measures.Approximation 1This approximation is within about 5% of the true value, so long asais not more than 3 times longer thanb(in other words, the ellipse is not too squashed):Approximation 2The famous Indian mathematicianRamanujancame up with this better approximation:Approximation 3Ramanujanalso came up with this one. First we calculate h:Then use it here:Infinite Series 1This is anexact formula, but it needs an infinite series of calculations to be exact, so in practice we still only get an approximation.First we calculatee(the eccentricity,notEulers number e):Then use this infinite sum formula:Which may look complicated, but expands like this:The terms continue on infinitely, and unfortunately we must calculate a lot of terms to get a reasonably close answer.Infinite Series 2But my favoriteexact formula(because it gives a very close answer after only a few terms) is as follows:First we calculate h:Then use this infinite sum formula:(Note: theis the Binomial Coefficientwith half-integerfactorials. wow!)It may look a bit scary, but it expands to this series of calculations:The more terms we calculate, the more accurate it becomes (the next term is 25h4/16384, which is getting quite small, and the next is 49h5/65536, then 441h6/1048576)ComparingJust for fun, I calculate the perimeter using the three approximation formulas, and the two exact formulas (but only thefirst four terms including the 1, so it is still just an approximation) for the following values ofaandb:CircleLinesa:1010101010b:105310Approx 1:62.83249.67346.38544.6544.429Approx 2:62.83248.44243.85740.60639.834Approx 3:62.83248.44243.85940.63939.984Series 1:62.83248.87645.17443.20442.951Series 2:62.83248.44243.85940.62339.884Exact*:2040* Exact: Whena=b, the ellipse is a circle, and the perimeter is2a(62.832. in our example). Whenb=0(the shape is really two lines back and forth) the perimeter is4a(40 in our example).They all get the perimeter of the circle correct, but onlyApprox 2 and 3andSeries 2get close to the value of 40 for the extreme case of b=0.Ellipse Perimeter Calculations ToolThis tool does the calculations from above, but with more terms for the Series.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=53 = 15一个的周缘圆圈被称为圆周：圆周= 2个半径周边公式三角形周长= A + B + C平方周长= 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- RR =半径扇区周长= R（+ 2）R =半径=在角度弧度椭圆周长=很辛苦！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。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：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（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！）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254/16384这是越来越相当小的，并且下一个是495/65536然后4416/1048576）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：圈行A：1010101010b：10五310约1：62.83249.67346.38544.6544.429约2：62.83248.44243.85740.60639.834约3：62.83248.44243.85940.63939.984系列一：62.83248.87645