Ellipse Circumference Calculator
Enter the two semi-axes, or switch modes and enter the full edge-to-edge width and length. Returns the oval's circumference using Ramanujan's second approximation, plus exact area and eccentricity.
Example: with How are you measuring? Semi-axes a, b (center to edge) · Semi-axis a (or full length) 10 · Semi-axis b (or full width) 6 → Circumference: 51.054 units.
- Area (exact)188.496 square units
- Eccentricity0.8 (0 = circle, near 1 = very flat)
- Semi-axes useda = 10, b = 6 (semi-axes)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Ramanujan's second approximation (1914) — the standard high-accuracy perimeter formula; no exact closed form exists.
Why ellipse perimeter needs an approximation
The area of an ellipse is beautifully simple — πab, exact. The circumference is not: it equals a so-called complete elliptic integral, which cannot be written in ordinary closed form. Mathematicians therefore use approximations, and the one Srinivasa Ramanujan published in 1914 — his second — is the standard: C ≈ π(a + b)[1 + 3h/(10 + √(4 − 3h))] with h = ((a − b)/(a + b))².
Its accuracy is remarkable. For the default 10 × 6 ellipse the error is far below the display rounding; even the worst case, an ellipse squashed completely flat, is off by only about 0.04% (it returns 3.9984a against the true 4a).
Semi-axes versus full width
The formula wants semi-axes: center to edge, like a radius. But an oval tabletop, mirror, or garden bed gets measured edge to edge. If your numbers are full length and width, switch to full mode and the calculator halves them before computing. Entering full measurements as semi-axes doubles the true perimeter — the most common error on this problem.
How it’s calculated
Ramanujan's second approximation: C ≈ π(a + b)[1 + 3h/(10 + √(4 − 3h))], where h = ((a − b)/(a + b))². Area = πab (exact); eccentricity e = √(1 − b²/a²) using the larger value as a. Full mode halves edge-to-edge inputs first. Rounded to 3 decimals.
The circumference is an approximation — worst-case error about 0.04% for a fully flattened ellipse and far smaller for everyday ovals; area and eccentricity are exact.
Example ovals
| Semi-axes a, b | Circumference | Note |
|---|---|---|
| 10, 6 | 51.054 | Default example |
| 8, 5 | 41.386 | Rounder oval |
| 12, 3 | 51.470 | Strongly elongated |
| 10, 10 | 62.832 | A circle — exactly 2πr |
Computed with Ramanujan's second approximation; the circle row matches 2πr exactly.
Common mistakes
- Entering full width and length as semi-axes — halve edge-to-edge measurements or use full mode.
- Using C = 2πr with an average radius; it noticeably underestimates elongated ovals.
- Confusing area (πab, exact) with circumference (approximate only).
- Assuming flattening an ellipse shrinks the perimeter toward zero — it approaches 4a, the back-and-forth length.
Frequently asked questions
What formula does this calculator use?
Ramanujan's second approximation: C ≈ π(a + b)[1 + 3h/(10 + √(4 − 3h))], where h = ((a − b)/(a + b))² and a, b are the semi-axes. It is the standard high-accuracy choice for ellipse perimeter.
Why is there no exact formula for an ellipse's circumference?
The exact answer is a complete elliptic integral of the second kind, which has no closed form in elementary functions. Area, by contrast, is exactly πab — the asymmetry that surprises most people.
My oval table is 60 by 40 inches. What do I enter?
Use full mode and enter 60 and 40 directly — the calculator halves them into semi-axes of 30 and 20. In semi-axes mode you would enter 30 and 20 yourself.
How accurate is the result?
Better than 0.04% even in the degenerate worst case (a completely flat ellipse), and effectively exact for realistic shapes — the error shrinks with the fifth power of h. Display rounding at 3 decimals dominates any formula error here.