Calculate all properties of an ellipse from semi-major and semi-minor axes. Includes area, circumference (Ramanujan), eccentricity, foci distance, directrix, latus rectum, and flattening.
An ellipse is the set of all points in a plane whose distances to two fixed points — the foci — sum to a constant. This elegant geometric definition underpins the most famous curves in nature and science, from the orbits of planets (Kepler's First Law states that every planet traces an ellipse with the Sun at one focus) to the cross-sections of cylinders cut at an angle.
Every ellipse is defined by two key measurements: the semi-major axis (a), which is half the longest diameter, and the semi-minor axis (b), which is half the shortest diameter. From these two values, every other property can be derived: area = πab, eccentricity e = √(1 − b²/a²), linear eccentricity (focal distance from center) c = √(a² − b²), the latus rectum ℓ = 2b²/a, and the directrices at ±a/e from the center.
Calculating the exact circumference of an ellipse is famously difficult — no closed-form expression exists in elementary functions. This calculator uses Ramanujan's remarkably accurate approximation: C ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))), where h = ((a − b)/(a + b))². For most practical purposes, this formula is accurate to within 0.001%.
Eccentricity ranges from 0 (a perfect circle) to values approaching 1 (an extremely elongated ellipse). Earth's orbit has e ≈ 0.0167 (nearly circular), while Halley's Comet has e ≈ 0.967 (extremely elongated). This calculator includes presets for planetary orbits and everyday objects so you can explore instantly.
The Ellipse Calculator — Area, Circumference, Eccentricity & Foci is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Area, Circumference (≈), Eccentricity (e) in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Area: A = πab Circumference (Ramanujan): C ≈ π(a+b)(1 + 3h/(10+√(4−3h))), h = ((a−b)/(a+b))² Eccentricity: e = √(1 − b²/a²) Linear eccentricity: c = √(a² − b²) Foci distance: 2c Latus rectum: ℓ = 2b²/a Directrix: d = a/e Flattening: f = 1 − b/a
Result: Area ≈ 188.4956 cm², Circumference ≈ 51.0543 cm, Eccentricity ≈ 0.8
For an ellipse with a = 10 cm, b = 6 cm: Area = π × 10 × 6 ≈ 188.50 cm². Eccentricity = √(1 − 36/100) = √0.64 = 0.8. Linear eccentricity c = √(100 − 36) = 8 cm. Foci are 16 cm apart. Latus rectum = 2 × 36/10 = 7.2 cm. Circumference ≈ 51.05 cm (Ramanujan).
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Ellipse Calculator — Area, Circumference, Eccentricity & Foci calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
A = πab, where a is the semi-major axis and b is the semi-minor axis. For a = 10 and b = 6, A = π × 10 × 6 ≈ 188.50 square units.
The exact circumference involves an elliptic integral — a special function with no closed-form expression in elementary terms. Approximations like Ramanujan's formula provide excellent accuracy for practical use.
Eccentricity (e) measures how "elongated" an ellipse is. e = 0 means a perfect circle, and values approaching 1 indicate an increasingly stretched shape. It's defined as e = √(1 − b²/a²).
The foci (singular: focus) are two special points inside the ellipse. The sum of distances from any point on the ellipse to both foci is constant (equals 2a). They lie on the major axis at distance c = ae from the center.
The latus rectum is the chord through a focus perpendicular to the major axis. Its length is ℓ = 2b²/a. It's important in orbital mechanics and optics.
Most planetary orbits are nearly circular. Earth: e ≈ 0.017, Mars: e ≈ 0.093, Mercury: e ≈ 0.206 (most eccentric planet). Comets can have eccentricities above 0.99.
An ellipse has two directrices — lines perpendicular to the major axis at distance a/e from the center. For any point on the ellipse, the ratio of its distance to a focus divided by its distance to the corresponding directrix equals e.