Conic Sections Calculator
Calculate properties of ellipses, hyperbolas, and parabolas from their equations.
Find foci, vertices, eccentricity, directrix, and semi-axes for any conic section.
What Are Conic Sections? Conic sections are the curves formed by intersecting a cone with a plane at different angles. They were first systematically studied by Apollonius of Perga, a Greek mathematician, around 200 BC in Alexandria, Egypt. The four types: circle (plane perpendicular to axis), ellipse (tilted cut), parabola (parallel to a slant), hyperbola (steeper than slant). Conics appear everywhere in physics: planetary orbits (ellipses), projectiles (parabolas), and spacecraft escape trajectories (hyperbolas).
Ellipse Standard form: x²/a² + y²/b² = 1 (a > b > 0, major axis along x-axis) Or: x²/b² + y²/a² = 1 (a > b > 0, major axis along y-axis) Semi-major axis: a. Semi-minor axis: b. Foci: c = √(a² − b²) apart from center. Eccentricity: e = c/a (0 < e < 1). Area = π × a × b. Perimeter ≈ π × [3(a+b) − √((3a+b)(a+3b))] (Ramanujan approximation). A circle is a special ellipse with a = b and e = 0.
Parabola Standard form: y = x²/(4p) or x = y²/(4p) p = distance from vertex to focus (= distance from vertex to directrix). Focus at (0, p). Directrix: y = −p (for upward parabola). Eccentricity: e = 1 (always). Parabolas focus parallel rays to a single point — used in telescopes, satellite dishes, and headlights.
Hyperbola Standard form: x²/a² − y²/b² = 1 (transverse axis along x) Foci: c = √(a² + b²). Eccentricity: e = c/a > 1. Asymptotes: y = ±(b/a)x. The difference of distances from any point to the two foci is constant = 2a. GPS and LORAN navigation use hyperbolic positioning — intersection of hyperbolas gives location.
Eccentricity Summary e = 0: circle. 0 < e < 1: ellipse. e = 1: parabola. e > 1: hyperbola. e = ∞: line (degenerate). Planetary orbit eccentricities: Earth = 0.017, Mars = 0.093, Mercury = 0.206, Pluto = 0.248, Halley’s Comet = 0.967.
General Equation Any conic: Ax² + Bxy + Cy² + Dx + Ey + F = 0. Discriminant B² − 4AC: < 0 → ellipse/circle; = 0 → parabola; > 0 → hyperbola.
Kepler’s Laws and Conics Kepler’s first law (1609): planets move in ellipses with the Sun at one focus. Open trajectories (escape or hyperbolic flyby): comets and spacecraft often travel on hyperbolic or parabolic paths relative to the Sun. The vis-viva equation gives orbital speed at any point: v² = GM(2/r − 1/a).