An ellipse is the closed curve traced by a point whose total distance from two fixed points (the foci) is constant. It is the general form of a circle: every circle is an ellipse with equal semi-axes. Ellipses appear throughout engineering, astronomy (planetary orbits), optics (ellipsoidal reflectors), architecture (elliptical arches), and everyday design — from oval mirrors to running tracks.
This calculator works out the area, perimeter, eccentricity, and focal distance of any ellipse in one step. It also runs in reverse: give it the area and one semi-axis and it will solve for the other.
The core formula
An ellipse has two semi-axes:
- a — half the length along the wider axis (semi-major)
- b — half the length along the narrower axis (semi-minor)
The area formula is exact and beautifully simple:
A = π × a × b
When a equals b the shape is a circle, and the formula collapses to π × r², confirming the special case. There is no similarly clean formula for the perimeter. This calculator uses Ramanujan’s second approximation (1914), which is accurate to better than one part in 10^15 for any eccentricity:
P ≈ π (a + b)(1 + 3h / (10 + sqrt(4 − 3h))), where h = (a − b)^2 / (a + b)^2
Solving for a missing semi-axis
Rearranging the area formula gives two further identities:
- a = A / (π × b) — when you know the area and the semi-minor axis
- b = A / (π × a) — when you know the area and the semi-major axis
Use the Solve for dropdown to switch modes. The highlighted result row is always the quantity that was calculated from the others.
Worked example
A landscaper is designing an oval flower bed that is 12 m wide and 8 m tall. What is the area?
- Semi-major axis: a = 12 / 2 = 6 m
- Semi-minor axis: b = 8 / 2 = 4 m
- Area = π × 6 × 4 = 24π ≈ 75.398 m²
Now suppose the landscaper knows the area must be exactly 50 m² and has fixed the width (a = 6 m). What height is needed?
b = 50 / (π × 6) = 50 / 18.849… ≈ 2.653 m, so the full height is about 5.305 m.
Switch the selector to “Semi-axis b”, enter a = 6 and Area = 50 to see the same result with the full working shown.
Eccentricity and focal distance
The focal distance is c = sqrt(a^2 − b^2). The two foci sit on the major axis at ±c from the centre. Eccentricity is e = c / a, ranging from 0 (perfect circle) to just below 1 (extremely elongated oval).
| Shape description | Eccentricity |
|---|---|
| Circle | 0 |
| Slightly oval | 0.1 – 0.3 |
| Moderately elongated | 0.5 – 0.7 |
| Very flat ellipse | 0.9+ |
Earth’s orbit has eccentricity ≈ 0.017 (nearly circular); Pluto’s is ≈ 0.25 (noticeably oval).
Precision and units
Select the number of decimal places (2 to 10) using the dropdown. The unit selector lets you label results in mm, cm, m, km, inches, feet, yards, or miles — no conversion is applied, so use consistent units for both inputs. Area is automatically labelled in the corresponding squared unit.
Every calculation runs entirely in your browser. No figures are uploaded or stored.