What is the formula for the Circumference of an ellipse?
Could it be (a+b)p when a and b represent axis of an ellipse?
Unfortunately there is no nice formula
for the circumference of an ellipse. The first guess of (a+b) x
pi is unfortunately not right. Mathematicians have invented
functions called elliptic functions to do things like this
[they're called elliptic because they were first discovered in
order to try and do this problem].
Before you say that inventing functions is a bit of a cheat
remember that functions like sine and cosine were invented to try
and solve problems to do with the circle. It is only because they
are easy to use and have been around for ages that they have
become regarded as common. It is possible that in a thousand
years elliptic functions may be so common that A-level students
could be being taught them!!!
If you are interested I can try and find out more about elliptic
functions... just let me know.
AlexB.
Can you not treat the ellipse as parametric equations and by integration find the length of the upper right quadrant:
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You will find that you can't do the
integration using functions that you know about. This integral is
in fact one of the original definitions of the elliptic
functions. Although nowadays they are defined in a more abstract
way.
AlexB.
Circumference of an ellipse:
===========================
It is true that there is no simple formula to compute
circumference of an ellipse. Indian Mathematician, S. Ramanujan
gave a wonderful formula which gives a very good approximate
value of circumference of an ellipse without using Integration or
Summation of Series.
Ramanujan's Formula:
t=(a-b)2/(a+b)2
| S=p(a+b)(1+3t/(10+ | ____ Ö4-3t | )) |