1、PerimeterPerimeter is the distance around a two-dimensional shape.Example: the perimeter of this rectangle is 7+3+7+3 = 20Example: the perimeter of this regular pentagon is 3+3+3+3+3 = 53 = 15The perimeter of a circle is called the circumference:Circumference = 2 radiusPerimeter FormulasTrianglePeri
2、meter = 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 in radiansPerimeter of an EllipseOn the Ellipse page we looked at the definition
3、 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 is very difficult to calculate!There are many formulas, here are some interesting ones. (Also see Calculation Tool below.)First
4、 Measure Your Ellipse!a and b are measured from the center, so they are like “radius“ measures.Approximation 1This 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 ellipse is not too “squashed“):Approximation 2The famous Ind
5、ian mathematician Ramanujan came up with this better approximation:Approximation 3Ramanujan also came up with this one. First we calculate “h“:Then use it here:Infinite Series 1This is an exact formula, but it needs an “infinite series“ of calculations to be exact, so in practice we still only get a
6、n approximation.First we calculate e (the “ eccentricity “, not Eulers 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 Se
7、ries 2But my favorite exact 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: the is the Binomial Coefficient with half-integer factorials . wow!)It may look a bit scary, but it expands to this serie
8、s 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 (
9、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 Linesa: 10 10 10 10 10b: 10 5 3 1 0Approx 1: 62.832 49.673 46.385 44.65 44.429Approx 2: 62.832 48.442 43.857 40.606 39.834Approx 3: 62.832 48.442 43.859 40.639 39.984Seri
10、es 1: 62.832 48.876 45.174 43.204 42.951Series 2: 62.832 48.442 43.859 40.623 39.884Exact*: 20 40* Exact: When a=b, the ellipse is a circle, and the perimeter is 2a (62.832. 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
11、the perimeter of the circle correct, but only Approx 2 and 3 and Series 2 get 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
12、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！
13、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
14、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 25 4 /16384 这是越来越相当小的，并且下一个是 49 5/65536 然后 441 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 行A： 10 10 10 10 10b： 10 五 3 1 0约 1： 62.832
15、 49.673 46.385 44.65 44.429约 2： 62.832 48.442 43.857 40.606 39.834约 3： 62.832 48.442 43.859 40.639 39.984系列一： 62.832 48.876 45.174 43.204 42.951系列二： 62.832 48.442 43.859 40.623 39.884精确*： 20 个 40*精确： 当 A = B 时，椭圆为圆形，且所述周边是 2 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。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。
