Ellipse calculator — area, exact perimeter, eccentricity and foci

How to use the ellipse calculator

1. Enter the Semi-major axis a, half the longest width.
2. Enter the Semi-minor axis b, half the shortest width.
3. Read Area (πab), then Perimeter (exact, AGM) and Perimeter (Ramanujan II).
4. Check Eccentricity, Focal distance from centre (c) and Semi-latus rectum.

What it does and when to use it

Ellipses turn up in oval tables, running tracks, mirrors, garden beds, arches, and the orbits of planets. The area tells you how much tabletop or turf you need. The perimeter gives the length of edging, trim or piping around an oval. Eccentricity and foci matter in physics and astronomy, for example when describing an orbit or the “whispering gallery” effect.

For a perfect circle, the circle calculator is simpler. The area calculator and perimeter calculator include the ellipse among many other shapes.

How it works

• Area: A = π × a × b
• Focal distance: c = √(a² − b²)
• Eccentricity: e = c ÷ a
• Semi-latus rectum: b² ÷ a (half the chord through a focus, at right angles to the long axis)
• Exact perimeter: P = (2π ÷ AGM(a, b)) × (a² − Σ 2ⁿ⁻¹ cₙ²), where AGM is the arithmetic-geometric mean of a and b, and cₙ is half the gap between the two means at each step. It converges in a handful of steps.
• Ramanujan’s second approximation: P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))), with h = ((a − b) ÷ (a + b))²

Worked examples

An oval table, step by step. Semi-axes 10 and 6 (a table 20 by 12 units).

1. Area: π × 10 × 6 = 60π = 188.4955592.
2. Focal distance: √(10² − 6²) = √64 = 8.
3. Eccentricity: 8 ÷ 10 = 0.8.
4. Semi-latus rectum: 6² ÷ 10 = 3.6.

The tool shows Area (πab): 188.4955592 and Focal distance from centre (c): 8. Both perimeter methods give 51.05399773, so about 51 units of edging.

Axes entered the other way round. a = 3 and b = 7. The tool swaps them and notes it. The exact perimeter is 32.68566675 and Perimeter (Ramanujan II): 32.68566665, so the approximation is off by only 0.0000001 for this fairly long ellipse. The eccentricity is 0.9035079029.

Limits and tips

• Both semi-axes must be greater than zero.
• The tool does not take the full axes, the foci or the eccentricity as inputs. Work back to a and b first.
• No unit menu: lengths are in your unit, area in that unit squared.
• The calculation runs in your browser.