Perimeter of an Ellipse

On the Ellipse page 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.

ellipse axes

Perimeter

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

There are many formulas, here are some interesting ones.

 

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

 

Approximation 1

This approximation will be 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"):

perimeter formula

Approximation 2

The famous Indian mathematician Ramanujan came up with this better approximation:

perimeter formula

Infinite Series 1

This is an exact formula, but it requires an "infinite series" of calculations to be exact, so in practice you still only get an approximation.

Firstly you must calculate e (the "eccentricity", not Euler's number "e"):

eccentricity formula

Then use this "infinite sum" formula:

perimeter formula

Which may look complicated, but expands like this:

perimeter formula

The terms continue on infinitely, and unfortunately you must calculate a lot of terms to get a reasonably close answer.

Infinite Series 2

But my favorite exact formula (because it gives a very close answer after only a few terms) is as follows:

Firstly you must calculate "h":

h formula

Then use this "infinite sum" formula:

h formula

(Note: the combinations-half-n is the Binomial Coefficient with half-integer factorials ... wow!)

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

perimter formula

The more terms you 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)

Comparing

Just for fun, I used the two approximation formulas, and the two exact formulas (but only the first four terms, so it is still just an approximation) to calculate the perimeter for the following values of a and b:

  a b Approx 1 Approx 2 Series 1 Series 2 Exact*
Circle 101062.83262.83262.83262.832 20π
  10549.67348.44248.87648.442  
  10346.38543.85745.17443.859  
  10144.6540.60643.20440.623  
Lines 10044.42939.83442.95139.884 40


* Exact:

  • When a=b, the ellipse is a circle, and the perimeter is 2πa (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 the perimeter of the circle correct, but only Approx 2 and Series 2 get close to the value of 40 for the extreme case of b=0.

My guess is that Infinite Series 2 is the best of them.