Ellipse Perimeter Calculator

Compare multiple approximation methods for the ellipse perimeter: Ramanujan I, Ramanujan II, infinite series, and more.

Enter Ellipse Dimensions

Results

Best Estimate (Infinite Series)
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units

Method Comparison

MethodResultError vs Series
Eccentricity (e)--
Parameter h--
Ratio b/a--

Step-by-Step: Ramanujan I

P = pi(3(a+b) - sqrt((3a+b)(a+3b)))

Ellipse Perimeter Approximation Methods

Unlike the circumference of a circle (C = 2*pi*r), there is no simple closed-form formula for the perimeter of a general ellipse. Over centuries, mathematicians have developed various approximation methods. This calculator computes and compares five popular methods side-by-side.

The Five Methods Compared

Ramanujan I (1914)

First approximation by Ramanujan. Simple, elegant, and accurate to about 0.02% for moderate eccentricities.

P = pi[3(a+b) - sqrt((3a+b)(a+3b))]

Ramanujan II (1914)

Second approximation by Ramanujan using parameter h. Even more accurate, especially for elongated ellipses.

P = pi(a+b)(1 + 3h/(10+sqrt(4-3h)))

Infinite Series

Converges to the exact value. Uses h = ((a-b)/(a+b))^2 and binomial coefficients.

P = pi(a+b) * SUM[C(0.5,n)^2 * h^n]

Euler's Formula

An early approximation. Less accurate but historically significant.

P = 2*pi*sqrt((a^2+b^2)/2)

Naive: pi(a+b)

The simplest approximation. Only accurate when a is close to b.

P = pi(a + b)

Understanding the h Parameter

Many advanced approximations use the parameter h = ((a-b)/(a+b))2. This parameter ranges from 0 (for a circle where a = b) to 1 (for a completely degenerate ellipse where b = 0). The parameter h is convenient because it stays small even when the eccentricity is moderate, making series expansions converge quickly.

Accuracy Comparison

For a typical ellipse with a = 10, b = 6 (eccentricity e = 0.8):

  • Naive pi(a+b): Error around 2-3%
  • Euler: Error around 0.5-1%
  • Ramanujan I: Error around 0.01-0.02%
  • Ramanujan II: Error around 0.0001%
  • Infinite Series (20 terms): Essentially exact for all practical purposes

Which Method Should You Use?

For most practical applications, Ramanujan I is more than sufficient. Its error is smaller than typical measurement uncertainty. For scientific computing or when dealing with very elongated ellipses, use Ramanujan II or the infinite series with 15-20 terms.

Historical Notes

The quest for an ellipse perimeter formula spans centuries. Isaac Newton worked on this problem in the 17th century. Leonhard Euler contributed his approximation in the 18th century. Colin Maclaurin and others developed series expansions. Ramanujan's two remarkable approximations from 1914 remain among the most useful formulas in mathematics, combining simplicity with extraordinary accuracy.

Mathematical Details

The exact perimeter is given by the complete elliptic integral of the second kind:

P = 4a * integral from 0 to pi/2 of sqrt(1 - e2 sin2(t)) dt

where e = sqrt(1 - b2/a2) is the eccentricity. This integral cannot be evaluated in closed form using elementary functions, which is why approximations are necessary for practical computation.

Tips for Using This Calculator

  • Increase the number of series terms for very elongated ellipses (large eccentricity) to get better accuracy.
  • For circles (a = b), all methods give the same exact result: 2*pi*r.
  • The error column shows absolute difference from the infinite series result (our best estimate).
  • The green-highlighted row in the comparison table indicates the method closest to the series result.