Enter the semi-major axis (a) and semi-minor axis (b) of your ellipse to calculate its circumference (perimeter). The Ellipse Circumference Calculator returns results using three formulas — Ramanujan II (most accurate approximation), Ramanujan I, and the Simple formula — along with the eccentricity and area of the ellipse.
Circumference (Ramanujan II)
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Circumference (Ramanujan I)
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Circumference (Simple Formula)
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Eccentricity
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Area
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The circumference of an ellipse is the total length of its boundary — equivalent to what 'perimeter' means for a polygon. Unlike a circle, an ellipse has no simple closed-form formula for its circumference, so mathematicians use approximations or infinite series to calculate it.
The exact circumference of an ellipse requires evaluating a complete elliptic integral of the second kind, which cannot be expressed in a finite number of elementary operations. This is why mathematicians like Ramanujan developed highly accurate approximation formulas instead.
Ramanujan's second approximation — p ≈ π(a+b)(1 + 3h/(10+√(4−3h))), where h = (a−b)²/(a+b)² — is widely considered the most accurate elementary approximation for the ellipse circumference. It is accurate to within a few parts per million for most ellipses.
The semi-major axis (a) is the longest radius of the ellipse — the distance from the center to the furthest point on the boundary. The semi-minor axis (b) is the shortest radius — the distance from the center to the nearest point. Together, they fully define the shape of the ellipse.
Eccentricity (e) measures how 'stretched' an ellipse is. It is calculated as e = √(1 − b²/a²), and ranges from 0 (a perfect circle) to just below 1 (a very elongated ellipse). A higher eccentricity means the ellipse is more elongated.
Yes — the term 'circumference' technically refers to the perimeter of a curved shape, so it applies to ellipses just as it does to circles. Both 'perimeter' and 'circumference' are used interchangeably when describing the boundary length of an ellipse.
When a = b, the ellipse becomes a perfect circle and the circumference formula simplifies to 2πa, which is the standard circle circumference formula. All three approximation formulas in this calculator will converge to the same exact value in that case.
Simply divide each diameter by 2 to get the semi-axes. The major diameter divided by 2 gives you the semi-major axis (a), and the minor diameter divided by 2 gives you the semi-minor axis (b). Then enter these values into the calculator.