How it’s calculated
An ellipse with semi-major axis a and semi-minor axis b has a simple area formula. Its perimeter has no elementary closed form: it equals 4a·E(e), a complete elliptic integral of the second kind, which we evaluate exactly with the fast arithmetic-geometric mean method. Ramanujan's approximation is shown for comparison.
Example: a = 5 ft, b = 3 ft gives Area = π × 15 ≈ 47.12 ft², c = √(25 − 9) = 4 ft, e = 0.8, foci 8 ft apart, and perimeter ≈ 25.53 ft (h = 1/16).
Frequently asked questions
How do I find the area of an oval?
Measure the longest and shortest widths, halve each to get a and b, then use A = πab. A 10 ft × 6 ft oval has area π × 5 × 3 ≈ 47.1 ft².
Why is there no simple formula for the perimeter?
The arc length of an ellipse is an elliptic integral, which cannot be written with elementary functions. This calculator computes it numerically to full precision.
What is eccentricity?
A measure of how stretched the ellipse is: e = √(1 − b²/a²). A circle has e = 0; a very flat ellipse approaches 1.
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Sources
- Precalculus 2e, 10.1 The Ellipse — OpenStax
- DLMF §19.8 Quadratic Transformations (arithmetic-geometric mean) — NIST Digital Library of Mathematical Functions
- DLMF §19.9 Inequalities (perimeter of an ellipse) — NIST Digital Library of Mathematical Functions
- Ellipse — Wolfram MathWorld
Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.
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