Ellipse Formulas
Perimeter ≈ π × [3(a+b) − √((3a+b)(a+3b))] (Ramanujan)
Eccentricity = √(1 − b²/a²) (where a ≥ b)
Where a = semi-major axis (half of the longest diameter), b = semi-minor axis (half of the shortest diameter).
Example
Ellipse with a = 8 cm, b = 5 cm
Area = π × 8 × 5 = 125.664 cm²
Perimeter ≈ π × [3(13) − √((24+5)(8+15))] = 41.602 cm (Ramanujan)
Eccentricity = √(1 − 25/64) = 0.781
Ellipse in Real Life
| Object | Notes |
|---|---|
| Planetary orbits | All planets orbit in ellipses (Kepler's 1st Law) |
| Whispering galleries | Sound from one focus reflects to the other |
| Oval sports tracks | Approximated as two semicircles + rectangle |
| Egg shape | Approximated by an ellipse |
| Camera lenses | Elliptical apertures used in cinematography |
Frequently Asked Questions
What is the difference between a circle and an ellipse?
A circle is a special case of an ellipse where the two axes are equal (a = b). The eccentricity of a circle is 0; as eccentricity approaches 1, the ellipse becomes more elongated (flatter).
Why is there no exact formula for the ellipse perimeter?
Unlike a circle, the arc length of an ellipse involves an elliptic integral — a type of integral with no simple closed-form solution. Ramanujan's formula is an approximation accurate to within 0.01% for most practical shapes.
What is eccentricity?
Eccentricity (e) measures how "stretched" an ellipse is. e = 0 is a perfect circle; e approaching 1 is a very elongated ellipse (nearly a line segment). Earth's orbital eccentricity is about 0.017 — nearly circular.