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Ellipse Circumference Calculator Ramanujan I & II + Eksak
Calculate keliling elips (circumference / perimeter of ellipse) from the semi-major axis a and semi-minor b. Compare aproksimasi Ramanujan I and Ramanujan II against the numerical exact exact value via the elliptic integral (AGM method) — complete with per-method error, bonus area πab, and SVG diagram.
Calculator
Eksak (AGM)
–
C = 4a·E(e), integral eliptik
Ramanujan II
–
error –
Ramanujan I
–
error –
Area (bonus)
–
A = πab
| Metode | Value | Selisih vs eksak | Error relatif |
|---|---|---|---|
| Eksak (AGM) | – | 0 | 0% |
| Ramanujan II | – | – | – |
| Ramanujan I | – | – | – |
Eksentrisitas e = – · Parameter h = ((a−b)/(a+b))² = –
Formulas used
Ramanujan I: C ≈ π·( 3(a+b) − √((3a+b)(a+3b)) )
Ramanujan II: C ≈ π(a+b)·( 1 + 3h/(10 + √(4−3h)) ), h = ((a−b)/(a+b))²
Eksak: C = 4a·E(e), e = √(1 − (b/a)²) — E(e) via AGM
Ramanujan II is almost always more accurate than Ramanujan I; both become exact when a = b (circle → 2πr).
Verified example
Each example is recomputed with high precision (60-digit Decimal, SciPy cross-verification). Click to fill the calculator.
1 · Lingkaran (sanity check)
a = b = 5
Eksak = 31.4159265359 = 2π×5.
All three methods agree, 0% error.
All three methods agree, 0% error.
2 · Elips umum
a = 10, b = 5
Eksak = 48.4422411027.
Ramanujan I error 0.000279951%; Ramanujan II error 0.000000045596%.
Ramanujan I error 0.000279951%; Ramanujan II error 0.000000045596%.
3 · Elips lonjong
a = 8, b = 3
Eksak = 36.3668627832.
Ramanujan I error 0.001876644%; Ramanujan II error 0.000001180842%.
Ramanujan I error 0.001876644%; Ramanujan II error 0.000001180842%.
4 · Elips jarum
a = 100, b = 1
Eksak = 400.1098329723.
Ramanujan I = 398.7413450574 (error 0.342028064%); Ramanujan II = 400.0142159433 (error 0.023897695349%).
Ramanujan I = 398.7413450574 (error 0.342028064%); Ramanujan II = 400.0142159433 (error 0.023897695349%).
Frequently asked questions (FAQ)
What is the formula for ellipse circumference?
There is no closed-form elementary formula for ellipse circumference. The two best approximations come from Ramanujan. Ramanujan I: C ≈ π(3(a+b) − √((3a+b)(a+3b))). Ramanujan II: C ≈ π(a+b)(1 + 3h/(10 + √(4−3h))) with h = ((a−b)/(a+b))². This calculator also computes the numerical exact value through the complete elliptic integral of the second kind E(e) using the AGM (arithmetic-geometric mean) method, so you can see how small the error of both approximations is.
Ramanujan I vs Ramanujan II — which is more accurate?
Ramanujan II is almost always more accurate. Verified example: for an ellipse with a=8 and b=3, the exact circumference is 36.3668627832. Ramanujan I gives 36.3661803067 (0.001876644% error), while Ramanujan II gives 36.3668623537 (only 0.000001180842% error) — about 1,500 times more accurate. For a near-circle ellipse (a=50, b=49) both approximations are practically identical to the exact value.
What does the exact value mean in this calculator?
The exact value is computed from C = 4a·E(e), where E(e) is the complete elliptic integral of the second kind and e = √(1 − (b/a)²) is the eccentricity. E(e) is computed with the AGM (arithmetic-geometric mean) method in 64-bit double precision, and the result has been cross-verified against SciPy’s ellipe function (Python) to within 1e-12. This is not a Ramanujan approximation — it is direct numerical integration from the definition.
Why is there no simple formula like the circle circumference (2πr)?
Ellipse circumference is defined by the integral ∫₀^{2π} √(a²sin²θ + b²cos²θ) dθ, which is an elliptic integral — proven impossible to express with elementary functions (polynomials, roots, trigonometry, exponentials). That is why mathematicians use approximations like Ramanujan’s or numerical series/integrals. When a = b = r, the integral collapses to 2πr, matching this calculator’s sanity check: a = b = 5 yields 31.4159265359 = 2π×5.
Are ellipse circumference and perimeter the same thing?
Yes, the same. Circumference and perimeter both mean the total length around the ellipse. This calculator answers both search terms with three methods: Ramanujan I, Ramanujan II, and numerical exact.
What if a = b (the ellipse becomes a circle)?
If a = b = r, the ellipse becomes a circle and all three methods in this calculator yield exactly 2πr. Verified example: a = b = 5 gives 31.4159265359 on all three methods (0% error). This is used as a sanity check that the formula implementation is correct.
When is the Ramanujan approximation good enough?
For ellipses that are not too elongated (low eccentricity), both Ramanujan approximations are practically perfect. Verified example: a = 50, b = 49 — both approximations match the exact value 311.0256060708 to 9 decimal places. Error only starts to show on very elongated ellipses: for a = 100, b = 1, Ramanujan I misses by 0.342028064% (398.7413450574 vs exact 400.1098329723), while Ramanujan II misses by only 0.023897695349% (400.0142159433).
What is the ellipse area shown as a bonus?
Ellipse area has a simple elementary formula: A = πab. Unlike the circumference, area needs no approximation. Verified example: a = 10, b = 5 gives area 157.079633. This calculator shows the area automatically next to the circumference as a bonus.
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