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How to approximate the perimeter of an ellipse?

It is easy to calculate the perimeter of the well-known special case of ellipse: the circle. For the circle, its perimeter is

P=2\pi R

where R is the radius of the circle.
In the rest of this article, we will consider an ellipse with a Cartesian equation:

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} = 1

Any translation or rotation does not change the value of the perimeter, so we have chosen to center this ellipse at the origin of the reference, aligned with the axes.

The classical approximation of the perimeter of the ellipse

The classic formula, which is found almost everywhere around the perimeter of the ellipse, is the following

P \approx 2\pi \sqrt{\dfrac{a^2+b^2}{2}}

Ramanujan's formulas for the perimeter of the ellipse

Ramanujan's first formula

Ramanujan has accustomed us to formulas, each one more incredible than the other, especially those on the number pi. Here is his first approximation of the perimeter of the ellipse.

P \approx \pi\left(3(a+b) - \sqrt{(3a+b)(a+3b)}\right)

Ramanujan's second formula

And here is his second formula, there we arrive on the incredible! We pose

h = \dfrac{(ab)^2}{(a+b)^2}

The approximation is then:

P \approx \pi (a+b) \left( 1 + \dfrac{3h}{1 + \sqrt{4-3h}}\right)

Nice huh? Fortunately we are not asked in class to learn such formulas!

Using an integral

We are going to establish an absolute formula but which does not admit a directly calculable value. The ellipse verifies the parametric equation

\left\{\begin{array}{lcl} x(t) & = & a \cos(t)\\ y(t) & = & b \sin(t) \end{array} \right., 0 \leq t \leq 2 \pi

We then have, according to a formula on the parametrized curves:

P = \int_0 ^{2\pi}\sqrt{x'(t)^2 +y'(t)^2 }dt

Which give

P = \int_0 ^{2\pi}\sqrt{a^2\sin^2(t) +b^2 \cos^2 (t)}dt

And thanks to various symmetries, we obtain:

P = 4\int_0 ^{\frac{\pi}{2}}\sqrt{a^2\sin^2(t)+b^2 \cos^2 (t)}dt

This integral has no explicit solution. One can then use methods of approximation of integrals to obtain again an approximation of this perimeter:

• Packages rectangle
• Trapezoid Formula
• Simpson's Formula
• Monte Carlo method

And to find out why this perimeter cannot be calculated, I recommend this video which will allow you to learn more!