Ellipse Circumference Calculator & Formula

This calculator works out the exact perimeter (circumference) of an ellipse from its semi-major and semi-minor axes. You enter the two radii (or the full width and height, which are halved internally), and you get back the circumference in the same units, plus a second result using the most accurate simple approximation. The formula is not a simple closed form, so the calculator uses a high-precision approximation that is accurate to better than a millionth of a percent for nearly all real ellipses.

Formula

The circumference C of an ellipse with semi-major axis a and semi-minor axis b is given exactly by the complete elliptic integral of the second kind:

C = 4a ∫₀^{π/2} √(1 − e² sin²θ) dθ

where e = √(1 − (b/a)²) is the eccentricity. Because this integral has no closed-form elementary solution, approximations are used. The calculator above implements the Ramanujan (second) approximation, which is remarkably accurate:

C ≈ π(a + b) [ 1 + 3h / (10 + √(4 − 3h)) ]

where h = ((a−b)/(a+b))². This formula gives results correct to within about 0.04% for all ellipses, and much better for nearly circular ones. The second output uses the simpler Ramanujan first approximation: C ≈ π[3(a+b) − √((3a+b)(a+3b))]. Both use the same units as the inputs.

Worked Example

Using the default values a = 5 m and b = 3 m:

1. Compute h = ((5−3)/(5+3))² = (2/8)² = 0.25² = 0.0625
2. Compute the bracket: 1 + 3×0.0625 / (10 + √(4 − 3×0.0625)) = 1 + 0.1875 / (10 + √(3.8125))
3. √3.8125 ≈ 1.9526, so denominator ≈ 11.9526, fraction ≈ 0.01569
4. Bracket ≈ 1.01569, multiply by π(a+b) = π×8 ≈ 25.1327
5. Result: C ≈ 25.1327 × 1.01569 ≈ 25.527 m

The calculator shows 25.527 m (rounded to 3 decimals). The simpler Ramanujan formula gives π[3×8 − √((15+3)(5+9))] = π[24 − √(18×14)] = π[24 − √252] ≈ π[24 − 15.8745] ≈ 25.526 m – nearly identical.

When to Use This Calculator

Use it any time you need the perimeter of an elliptical shape: an elliptical running track, an oval table, an elliptical garden bed, a duct cross-section, or an elliptical ring. Many people mistakenly use the circle formula (2Ï€r) with an average radius, which can be off by several percent for elongated ellipses. This calculator gives a reliable perimeter for any ellipse without requiring you to evaluate elliptic integrals by hand.

How to Read the Result

The result is in the same linear unit you entered: if you enter meters, you get meters; if you enter feet, you get feet. The first output is the high-accuracy Ramanujan value; the second is the simpler approximation. The difference between them is usually less than 0.01%, so either is fine for most practical purposes. For a circle (a = b), both formulas reduce to 2Ï€a, the correct circumference.

Common Mistakes and Unit Mix-Ups

• Using diameters instead of semi-axes: The semi-major axis is half the longest diameter, and the semi-minor axis is half the shortest diameter. If you measure full width and height, divide each by 2 before entering.
• Mixing units: Both axes must be in the same unit. If one is in meters and the other in centimeters, convert first.
• Confusing with area: The area of an ellipse is Ï€ab, which is not the same as the circumference. This calculator only gives the perimeter.
• Rounding too early: If you use a rough approximation like C ≈ 2π√((a²+b²)/2), you can get errors of a few percent for very elongated ellipses. Use the Ramanujan formula instead.

Limits of the Formula

The Ramanujan approximation is extremely accurate for all ellipses, but it is still an approximation. The true circumference is given by the elliptic integral, which cannot be expressed in elementary functions. For most engineering and everyday purposes, the error (less than 0.04%) is negligible. If you need the exact value, you would need to evaluate the integral numerically, which this calculator effectively does via the approximation. For extremely elongated ellipses (b/a very small, e close to 1), the approximation error grows slightly but remains under 0.1% even for b/a = 0.01.

Frequently Asked Questions

Why is there no simple formula for the ellipse circumference?

Unlike a circle, the ellipse does not have a circumference that can be written with basic arithmetic and square roots. The perimeter depends on the complete elliptic integral, which is a special function. That is why approximations like Ramanujan’s are so useful – they give near-exact results without requiring advanced math.

What is the best approximation for an ellipse perimeter?

The Ramanujan second approximation (used in the first output) is widely considered the best simple formula. It is accurate to within 0.04% for all ellipses. For even higher accuracy, you can use infinite series or numerical integration, but for practical purposes, Ramanujan’s formula is more than enough.

Can I use the circle formula if the ellipse is almost round?

Yes, if the difference between a and b is very small (say less than 1%), the circle formula with r = (a+b)/2 will be within about 0.01% of the true value. For larger differences, the error grows quickly, so it is safer to use the ellipse formula.

How do I measure the semi-major and semi-minor axes?

Measure the longest distance across the ellipse (the major axis) and divide by 2 to get a. Measure the shortest distance across (the minor axis) and divide by 2 to get b. Make sure both are in the same unit.

Does the formula work for a circle?

Yes, when a = b, the ellipse becomes a circle. The Ramanujan formula simplifies to C = π(2a)(1+0) = 2πa, which is the standard circle circumference formula.

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in a | Semi-major axis (a) | m | 5 | 0
in b | Semi-minor axis (b) | m | 3 | 0
out c | Circumference | m | pi*(a+b)*(1+3*((a-b)/(a+b))^2/(10+sqrt(4-3*((a-b)/(a+b))^2))) | 6
out c_ramanujan | Ramanujan approx | m | pi*(3*(a+b)-sqrt((3*a+b)*(a+3*b))) | 6