Ellipse Calculator: Area, Perimeter (Ramanujan) and Eccentricity
Enter the two semi-axes (half the longest and half the shortest width). You get the area, a close perimeter estimate and the conic values used in Class 11.
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Area
188.4956cm²
- Perimeter
- 51.054 cm
- Eccentricity (e)
- 0.8
- Centre to focus (c)
- 8 cm
- Latus rectum
- 7.2 cm
An ellipse has no exact simple perimeter formula. Ramanujan's approximation is accurate to a tiny fraction of a percent for most shapes.
Show working
- A = πab = π × 10 × 6 = 188.4956 cm²
- P ≈ π[3(a + b) − √((3a + b)(a + 3b))] = 51.054 cm
- P (Ramanujan II) ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))), h = ((a − b)/(a + b))² → 51.054 cm
- c = √(a² − b²) = 8 cm; e = c / a = 0.8
How to use
- Enter the semi-major axis a and semi-minor axis b.
- Pick a unit (and π under "More options" if needed).
- Read the area, perimeter and eccentricity.
Good to know
Area = πab is exact. The perimeter has no simple exact formula, so the calculator uses Ramanujan's approximation P ≈ π[3(a + b) − √((3a + b)(a + 3b))], and shows his second, even closer formula in the steps.
From NCERT Class 11 "Conic Sections": c = √(a² − b²) is the distance from the centre to each focus, eccentricity e = c/a (0 for a circle, close to 1 for a long thin ellipse) and the latus rectum is 2b²/a.
Enter half-widths, not full widths. If you measured an oval table as 160 × 100 cm, enter a = 80 and b = 50. The larger value is always taken as a.
Questions
What is the area of an ellipse?
πab, where a and b are the semi-axes. With a = 10 and b = 6 it is 60π ≈ 188.5.
Why is the perimeter only approximate?
The exact perimeter needs an elliptic integral. Ramanujan's formula is accurate enough for practical work and exam use.
What is eccentricity?
A measure of how stretched the ellipse is: e = √(1 − b²/a²). A circle has e = 0.
Is a circle an ellipse?
Yes, one with a = b. The calculator then gives the usual πr² and 2πr.
Is anything I enter stored?
No. The calculation runs in your browser and nothing is sent to a server.