Ellipse Equation Calculator — Foci, Eccentricity & Area
Calculate ellipse properties — area, perimeter, foci, eccentricity, semi-major/minor axes — from standard equation parameters.
Ellipse Guide
Ellipse Equation
Standard form (centred at origin): x²/a² + y²/b² = 1. Where a = semi-major axis, b = semi-minor axis (with a > b). Centred at (h, k): (x − h)²/a² + (y − k)²/b² = 1. Foci location: distance c from centre along major axis. c = √(a² − b²). For an ellipse with horizontal major axis: foci at (h ± c, k). For vertical major axis: (h, k ± c). Eccentricity: e = c/a. Always between 0 and 1 for an ellipse. e = 0: perfect circle (a = b). e → 1: increasingly elongated. e > 1: hyperbola (different curve).
Area and Perimeter
Area: A = π × a × b. Simple formula. For circle (a = b = r): A = πr². ✓ Example: ellipse with a=5, b=3. A = π × 5 × 3 = 47.12 sq units. Perimeter: no closed-form solution exists. Numerous approximations: Ramanujan (1914): P ≈ π[3(a+b) − √((3a+b)(a+3b))]. Very accurate even for eccentric ellipses. Approximation 1: P ≈ 2π√((a² + b²)/2). Less accurate. Exact value: requires elliptic integral E(e). Cannot be expressed in elementary functions. This was a major problem in 18th-19th century mathematics
Foci Property
Reflective property: any line from one focus reflects off the ellipse to pass through the other focus. Whispering galleries: in elliptical rooms (St Paul's Cathedral Whispering Gallery, Statuary Hall in US Capitol), whispers at one focus are clearly heard at the other focus, even from far away. Lithotripsy: medical device to break kidney stones. Generates shock wave at one focus, focuses energy on the stone (at the other focus) inside the body. Solar concentrators: parabolic and elliptical mirro
Astronomical Importance
Kepler's First Law: planets orbit the Sun in ellipses with the Sun at one focus. Not the centre — the focus. Earth's orbit: eccentricity 0.0167 (almost circular). Distance to Sun varies 147-152 million km. Pluto's orbit: e = 0.249 (more eccentric). Comets: highly eccentric. Halley's Comet: e = 0.97. Long elliptical orbits taking it from inside Mars to past Neptune. Spacecraft: most use elliptical transfer orbits. Hohmann transfer: most efficient way to move between two circular orbits. Geostatio
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