This conic sections calculator identifies and fully describes the four conic families — circle, ellipse, hyperbola and parabola — from their standard-form parameters: center/vertex, axes, foci, eccentricity and key lines.
Four Curves, One Slicing Cone — e Sorts Them All
The circle member of the family gets its own deep dive in our Circle Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- Ellipse a = 5, b = 3 (defaults): c = √(25 − 9) = 4 — foci (±4, 0), e = 0.8, area 15π ≈ 47.12; the 3-4-5 triangle hiding in every textbook ellipse
- Hyperbola a = 4, b = 3: c = √(16 + 9) = 5, e = 1.25, asymptotes y = ±(3/4)x — the same triple, addition instead of subtraction
- Parabola (y)² = 8x: 4p = 8 → p = 2 (enter a = 2) — focus (2, 0), directrix x = −2, latus rectum 8
Eccentricity: The Family Sorter
- e = 0: circle (a special ellipse with coincident foci).
- 0 < e < 1: ellipse — orbits of planets live here (Earth: e ≈ 0.017, nearly circular).
- e = 1: parabola — the escape-velocity boundary curve.
- e > 1: hyperbola — flyby trajectories and cooling-tower silhouettes.
Related Tools
- Circle Calculator — the e = 0 member in depth
- Domain and Range Calculator — function sets
- Math Calculators — the full collection
Frequently Asked Questions
What are the four conic sections?
Circle, ellipse, parabola and hyperbola — the curves from slicing a double cone with a plane at different angles. Each has a standard equation, a focus (or two) and an eccentricity band: circle e = 0, ellipse e < 1, parabola e = 1, hyperbola e > 1. This calculator describes each from its standard-form parameters.
How do you find the foci of an ellipse?
With c = √(a² − b²) along the major axis from the center: a = 5, b = 3 → c = 4, foci at (±4, 0) for a centered horizontal ellipse. For hyperbolas the relation flips to c = √(a² + b²) — the sign difference that separates the families.
What is eccentricity?
e = c/a — how far the foci sit from the center relative to the size. It measures departure from circularity: Earth’s orbit e ≈ 0.017 is nearly round; Halley’s comet e ≈ 0.967 is a stretched ellipse nearly at the parabolic edge.
How do I read a parabola’s focus from its equation?
For (y − k)² = 4p(x − h): vertex (h, k), focus p units inside the opening at (h + p, k), directrix the line x = h − p. y² = 8x means 4p = 8, p = 2: focus (2, 0). The 4p coefficient bundling fools everyone once — this tool takes p directly.
What are a hyperbola’s asymptotes?
y − k = ±(b/a)(x − h) for the horizontal form — the two lines the branches approach forever. With a = 4, b = 3 they’re y = ±0.75x (centered). Swap a/b roles for vertical hyperbolas.
Is this conic sections calculator free?
Yes — free, instant, no sign-up.