How to use
- Enter semi-axis a.
- Enter semi-axis b.
- Read area and approximate perimeter.
Formula
Area is π × a × b. There is no simple closed form for the exact perimeter, so the page uses Ramanujan’s approximation: π × (a + b) × (1 + 3h ÷ (10 + √(4 − 3h))), where h is ((a − b) ÷ (a + b))². Eccentricity is √(1 − (smaller ÷ larger)²). Results are rounded to four decimal places.
Worked example
The fields open at a = 5 and b = 3. Area is π × 5 × 3 = 47.1239. The approximate perimeter is 25.527. Eccentricity is 0.8. If both semi-axes were 5, the area would be the circle π × 25 and the eccentricity would be 0.
How to read the result
47.1239 is the area inside the ellipse. 25.527 is an approximation of the outline, not an exact length. The semi-axes are half the full width and half the full height. A full width of 10 and a full height of 6 are the same ellipse as semi-axes 5 and 3.
When this does not apply
Do not type the full width into a semi-axis box. That would describe an ellipse twice as large in that direction. The perimeter is not exact. A circle calculator is the better page when the two axes are equal and you want an exact circumference.