Core answer: Ellipse area = πab, where a and b are the semi-major and semi-minor axes (half-lengths of the long and short diameters). A 20 × 12 cm oval table has semi-axes 10 and 6, so area = π×60 ≈ 188 cm²… wait, that's 0.019 m²; a 2.0 × 1.2 m table gives π×1×0.6 ≈ 1.88 m². Circumference has NO simple exact formula — Ramanujan's approximation is the practical standard.
The two axes and the area
An ellipse is a circle stretched in one direction: semi-major axis a (half the long way), semi-minor b (half the short way). Stretching a circle of radius a by factor b/a scales its area by b/a: A = πab. At a = b the ellipse is a circle and πab = πr² ✓.
Circumference: no elementary formula (use Ramanujan)
C ≈ π[3(a+b) − √((3a+b)(a+3b))] — accurate to ~10⁻⁹ for typical shapes.
| Shape | a / b | Ramanujan C | Area πab |
|---|---|---|---|
| Nearly round | 10 / 9.5 | ≈ 61.3 | 298 |
| 2:1 oval | 10 / 5 | ≈ 48.4 | 157 |
| Very flat | 10 / 1 | ≈ 38.8 | 31.4 |
Simple √((a²+b²)/2) × 2π underestimates for flat ellipses; prefer Ramanujan.
Worked examples
Example 1 — Oval running track question. A table 1.8 × 0.9 m (a=0.9, b=0.45): area π×0.405 ≈ 1.27 m²; edging Ramanujan ≈ π[3×1.35 − √((3.15)(2.25))] ≈ π[4.05 − 2.66] ≈ 4.37 m of trim.
Example 2 — Planetary orbits. Earth orbits in an ellipse of a = 149.6 M km, b ≈ 149.58 M km — so nearly circular that πab vs a circle differs by <0.03%; seasons come from axial tilt, not distance.
Example 3 — Whispering gallery. Sound from one focus reflects to the other focus — the ellipse's reflection property. Stand at one focus of an elliptical dome and whisper; your friend at the other focus hears you across the hall.
Eccentricity: how "stretched"
e = √(1 − b²/a²), from 0 (circle) toward 1 (needle). Halley's comet: e ≈ 0.967 — a dramatic sliver; Earth's orbit: e ≈ 0.0167 — visually indistinguishable from a circle.
Common mistakes and myths
- Using full diameters in πab — a and b are SEMI-axes (halves); using full lengths quadruples the area.
- Believing π(a+b)/2 or 2π√(ab) is exact — no closed form exists in elementary functions; Ramanujan is the practical answer.
- "Orbits are very elliptical" — most planets are nearly circular; textbooks exaggerate eccentricity for visibility.
- Confusing foci with center — foci sit at ±c from center where c² = a² − b²; the sun is at a focus (Kepler's first law), not the center.
- Assuming ellipse area needs calculus — πab is exact and elementary; only the circumference resists.