Ellipse Calculator

Enter semi-axes a and b. Area is exactly πab. Perimeter uses Ramanujan's closed-form approximation, not an elliptic-integral solver.

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How to calculate ellipse area and perimeter

Enter semi-axis a along one principal direction. The default a is 2.
Enter semi-axis b along the other principal direction. The default b is 1.
Read exact area as π × a × b using this browser's Math.PI.
Read perimeter from Ramanujan's approximation. Treat it as a high-quality estimate, not an elliptic-integral exact value.
When a equals b, the figure is a circle and area becomes πa²; use the circle page if you prefer radius-only inputs.

Ellipse area and perimeter

Plane oval defined by two perpendicular semi-axes

Exact area	A = π a b
Perimeter method	Ramanujan approximation (not elliptic integrals)
Default example	a = 2, b = 1 give area 2π
Circle limit	When a = b the ellipse is a circle of radius a
Axes meaning	a and b are half the major and minor diameters
Privacy	Arithmetic runs in this tab. CZNull does not receive the numbers.

What an ellipse is on this page

An ellipse is the set of plane points whose distances to two foci sum to a constant. This calculator does not ask for foci. It asks for semi-axes a and b, which are half the widths along the two symmetry axes. Defaults a = 2 and b = 1 describe a 4-by-2 bounding box.

Exact area πab

Area is exactly A = π a b. That identity is closed-form. For a = 2 and b = 1, area equals 2π. The page uses IEEE π from Math.PI in this browser. Swapping a and b does not change area.

Ramanujan perimeter approximation

The true ellipse circumference is a complete elliptic integral of the second kind. Ramanujan gave a compact approximation: let h = ((a − b)²) / ((a + b)²), then P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))). The engine reports that estimate and labels it as approximate. Error grows for extremely elongated ellipses but stays tiny for everyday a and b ratios.

Use cases

Oval tracks, planetary orbit sketches that use semi-major and semi-minor axes, and design layouts where you know both half-widths. Orbital mechanics that need eccentricity and focus distance are outside this form.

Limits

No elliptic-integral exact perimeter. No 3D ellipsoid volume. Semi-axes must be positive. See the disclaimer.

Ellipse calculator questions

Is ellipse area approximate on this page?

No. Area uses the exact formula A = π a b. Only the perimeter row is approximate.

Why approximate the perimeter at all?

The exact circumference needs elliptic integrals. Ramanujan's formula is a fast closed form that is accurate enough for most homework and design checks.

What is Ramanujan's h term?

h = ((a − b)²) / ((a + b)²). It measures how far the axes differ. When a = b, h = 0 and the approximation collapses to the circle circumference 2πa.

Do a and b need a major/minor order?

Area does not care about order. Perimeter approximation also accepts either order. Label the longer half-width as the semi-major axis in your own notes if you need that convention.

What area do a = 2 and b = 1 give?

Exactly 2π using this browser's π.

Is an ellipsoid the same tool?

No. This page is a 2D ellipse. A 3D ellipsoid needs three semi-axes and a different formula set.