About

An ellipse is a conic section defined by two focal points where the sum of distances from any point on the curve to both foci remains constant. Its area depends on two parameters: the semi-major axis a and the semi-minor axis b. Miscalculating these values propagates errors into material estimates, land surveys, and engineering tolerances. A circle is merely a degenerate ellipse where a = b. This calculator computes exact area via A = πab, Ramanujan's perimeter approximation, and orbital eccentricity. It assumes a perfect geometric ellipse. Real-world ellipses in machining or architecture may deviate due to manufacturing tolerances or material deformation.

Formulas

The area enclosed by an ellipse is computed from its two semi-axes:

A = π ⋅ a ⋅ b

where A is the enclosed area, a is the semi-major axis (longest radius), b is the semi-minor axis (shortest radius), and π ≈ 3.14159265.

No closed-form expression exists for the perimeter of a general ellipse. Ramanujan's second approximation provides accuracy within 0.005% for most cases:

P ≈ π ⋅ (3(a + b) − √(3a + b)(a + 3b))

where P is the approximate perimeter (circumference) of the ellipse.

Eccentricity quantifies deviation from a circle. It ranges from 0 (circle) to 1 (degenerate line):

e = √1 − b²a²

where e is the eccentricity, valid when a ≥ b. The linear eccentricity (focal distance from center) is c = √a² − b².

Reference Data

Real-World Ellipse	Semi-Major Axis a	Semi-Minor Axis b	Eccentricity e	Area
Circle (degenerate)	5 m	5 m	0.000	78.54 m²
Running Track (inner)	63.41 m	36.50 m	0.818	7,270 m²
Earth's Orbit	149.60 Gm	149.58 Gm	0.0167	7.03 × 10²² km²
Mars's Orbit	227.94 Gm	226.94 Gm	0.0934	1.63 × 10²³ km²
Pluto's Orbit	5,906.4 Gm	5,720.6 Gm	0.2488	1.06 × 10²⁶ km²
Halley's Comet	2,667.9 Gm	676.8 Gm	0.9671	5.67 × 10²⁴ km²
Whispering Gallery (St Paul's)	15.25 m	15.25 m	≈ 0	730.6 m²
Egg (average chicken)	2.8 cm	2.2 cm	0.619	19.35 cm²
Football (American, cross)	14.0 cm	8.6 cm	0.789	378.3 cm²
Elliptical Mirror (telescope)	0.50 m	0.35 m	0.714	0.5498 m²
Satellite Dish (oval)	1.20 m	0.90 m	0.661	3.393 m²
Elliptical Pool	8.00 m	4.50 m	0.827	113.1 m²
Piazza del Campidoglio	27.0 m	22.0 m	0.580	1,866 m²
Mercury's Orbit	57.91 Gm	56.67 Gm	0.2056	1.03 × 10²² km²
Kepler-16b Orbit	104.6 Gm	104.5 Gm	0.0069	3.43 × 10²² km²

Frequently Asked Questions

A circle is a special case of an ellipse where the semi-major axis a equals the semi-minor axis b, both equal to the radius r. Substituting into A = πab yields A = πr², which is the standard circle area formula. The ellipse formula generalizes the circle formula to two independent radii.

The area integral of an ellipse resolves to the closed form πab. The perimeter integral, however, produces a complete elliptic integral of the second kind, which has no elementary closed-form solution. Ramanujan's approximation used here achieves error below 0.005% for eccentricities under 0.95. For extreme eccentricities approaching 1, numerical integration methods like Gauss-Kuzmin are preferred.

By convention, the semi-major axis a is always the longer axis. If you enter b > a, this calculator automatically swaps the values internally so that a ≥ b before computing eccentricity. The area result πab remains identical regardless of assignment since multiplication is commutative.

At e = 0 the ellipse is a perfect circle. Earth's orbital eccentricity of 0.0167 makes it nearly circular. At e = 0.5, the major axis is about 15% longer than the minor. Halley's Comet at e ≈ 0.967 traces an extremely elongated path. As e approaches 1, the ellipse degenerates into a line segment.

Yes. JavaScript uses IEEE 754 double-precision floats, accurate to about 15 significant digits. This supports axes from 10⁻³⁰⁸ to 10³⁰⁸. For astronomical orbits (gigameters) or nanotechnology (nanometers), results remain precise within floating-point limits. The unit selector lets you work in the most natural scale for your domain.

Measure the longest diameter across the ellipse and divide by 2 to get a. Then measure the shortest diameter perpendicular to the first, through the center, and divide by 2 for b. For irregular objects, take multiple measurements and average. Pro tip: for an elliptical garden bed or pool, use string and two stakes at the foci. The total string length equals 2a.