Ellipse Calculator
Calculate the area, circumference, and eccentricity of an ellipse from semi-major and semi-minor axes. Includes foci distance, directrix, latus rectum, and bounding box.
Area
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Circumference (approx) —
Eccentricity —
Extended More scenarios, charts & detailed breakdown ▾
Area
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Circumference —
Eccentricity —
Professional Full parameters & maximum detail ▾
Eccentricity & Focus
Linear Eccentricity (c) —
Directrix Distance —
Latus Rectum —
Bounding Areas
Inscribed Rectangle Area —
Bounding Box Area —
How to Use This Calculator
- Enter the Semi-Major Axis (a) — the larger half-diameter.
- Enter the Semi-Minor Axis (b) — the smaller half-diameter.
- Get area, circumference (Ramanujan approximation), and eccentricity.
- Use the Foci tab to find the distance between foci.
- The Professional tab adds directrix, latus rectum, and bounding box area.
Formula
Area = πab | Eccentricity = c/a where c = √(a²−b²)
Circumference (Ramanujan) = π(a+b)[1 + 3h/(10+√(4−3h))], h = (a−b)²/(a+b)²
Example
a=5, b=3: Area = π×5×3 ≈ 47.12 units², c = √(25−9) = 4, e = 4/5 = 0.8.
Frequently Asked Questions
- The area of an ellipse is A = π × a × b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). For a = 6 and b = 4: A = π × 6 × 4 ≈ 75.40 square units. When a = b = r, the ellipse becomes a circle and the formula reduces to A = πr² as expected. Area grows linearly with each axis — doubling a doubles the area; doubling both a and b quadruples it.
- Unlike a circle, an ellipse has no simple closed-form circumference formula. The most widely used accurate approximation is the Ramanujan formula: C ≈ π(a+b)[1 + 3h/(10+√(4−3h))], where h = (a−b)²/(a+b)². For a circle (a=b), h=0 and C = π(a+b) = 2πa. For example, a=5, b=3: h = (2/8)² = 0.0625, then C ≈ π(8)[1 + 3×0.0625/(10+√(4−0.1875))] ≈ 25.53 units. The exact answer requires an elliptic integral.
- Eccentricity measures how "stretched" an ellipse is, compared to a perfect circle. Formula: e = c/a, where c = √(a² − b²) is the focal distance and a is the semi-major axis. Eccentricity ranges from 0 (perfect circle, b = a) to just below 1 (extremely elongated ellipse). For a = 5, b = 3: c = √(25−9) = 4, e = 4/5 = 0.8 — fairly elongated. Earth's orbit has eccentricity ≈ 0.0167 (nearly circular). A comet's elliptical orbit can have e close to 1.
- The foci (singular: focus) are two special fixed points inside an ellipse. Each focus is at distance c = √(a² − b²) from the center along the major axis. The defining property of an ellipse: for any point P on the ellipse, the sum of distances from P to both foci equals 2a (the major diameter). For a = 5, b = 3: c = 4, so the foci are at (±4, 0) from the center. Ellipses with both foci at the same point are circles. Elliptical orbits have one focus at the body being orbited (Kepler's first law).
- The latus rectum is the chord (line segment) that passes through a focus and is perpendicular to the major axis (parallel to the minor axis). Its length is 2b²/a. For a = 5, b = 3: latus rectum = 2 × 9 / 5 = 3.6 units. Half the latus rectum (b²/a) is called the semi-latus rectum and appears in orbital mechanics formulas. In a circle (a = b = r), the latus rectum = 2r (the diameter), as expected. The latus rectum describes the "width" of an ellipse at the focus.