Ellipse Circumference Calculator
Calculate the circumference (perimeter) of an ellipse from its semi-major and semi-minor axes using the highly accurate Ramanujan approximation. Unlike circles, ellipses have no simple exact perimeter formula, so this calculator uses the best-known closed-form approximation to deliver results accurate to several decimal places for any ellipse shape.
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Formula
How it works: This uses Ramanujan's second approximation, where h = ((a - b)^2) / ((a + b)^2). The semi-major axis (a) is the longer half-diameter and the semi-minor axis (b) is the shorter one. The formula is extremely accurate for all ellipses, with error typically less than 0.01% even for very elongated shapes.
Worked Example
Frequently Asked Questions
Why is the ellipse circumference formula an approximation?
Unlike circles, there is no simple closed-form formula for the exact perimeter of an ellipse using basic mathematical functions. The exact answer requires an elliptic integral, which cannot be expressed in elementary terms. Ramanujan's approximation is accurate to many decimal places for practical purposes.
Does it matter which axis is entered as a or b?
No. The calculator automatically handles axis ordering. Whether you enter the longer axis as a or b, the result will be the same. The formula uses the absolute difference between the axes.
How accurate is the Ramanujan approximation?
Extremely accurate. For most practical ellipses, the error is less than 0.01%. Even for very elongated ellipses (where one axis is many times larger than the other), the approximation remains within 0.1% of the exact value.
What happens when both axes are equal?
When a = b, the ellipse becomes a circle. The formula correctly reduces to C = 2 x pi x r, confirming the circle is a special case of the ellipse.
What are real-world uses for ellipse circumference?
Calculating the perimeter of oval running tracks, designing elliptical mirrors and lenses, determining the orbital circumference of planets (which follow elliptical orbits), and sizing oval frames, gaskets, and decorative borders.
How can I put this Ellipse Circumference Calculator on my blog or website?
Yes, the Ellipse Circumference Calculator is fully embeddable. Tap "Embed" above to configure appearance and copy the code. It is free to use, works on any platform (HTML, WordPress, CMS), and adjusts to any screen size automatically. Visit calculory.com/services/embed-calculators for the complete guide.
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