Conic Sections Calculator — Circle, Ellipse, Parabola, Hyperbola
Select a conic section type — circle, ellipse, parabola or hyperbola — enter its parameters and instantly get the standard equation, key properties (foci, eccentricity, asymptotes) and a plotted curve.
Conic type
units
- 1
Circumference (2πr)
2 × π × 5 = 31.4159 - 2
Area (πr²)
π × 5² = 78.5398
How does this calculator work?
All conic sections are slices of a double cone. A circle (x²+y²=r², e=0), ellipse (x²/a²+y²/b²=1, e<1), parabola (y=ax², e=1) and hyperbola (x²/a²−y²/b²=1, e>1) differ only in eccentricity. Select a type to compute its equation, foci, asymptotes and a plotted curve.
Formula
How this is calculated
All four conic sections arise by slicing a double cone with a plane at different angles. The eccentricity e classifies them: a circle has e = 0 (a degenerate ellipse where both foci coincide at the centre); an ellipse has 0 < e < 1; a parabola has e = 1; a hyperbola has e > 1.
For a circle of radius r: circumference = 2πr, area = πr². For an ellipse with semi-major axis a and semi-minor axis b (a ≥ b): the focal distance is c = √(a² − b²), eccentricity e = c/a, area = π·a·b, and the perimeter is approximated by Ramanujan's formula π(a+b)(1 + 3h/(10+√(4−3h))) where h = ((a−b)/(a+b))². For a parabola y = ax²: the vertex is at the origin, the focus is at (0, 1/(4a)) and the directrix is the horizontal line y = −1/(4a). For a hyperbola x²/a² − y²/b²= 1: the focal distance is c = √(a² + b²), eccentricity e = c/a > 1, and the two asymptotes are y = ±(b/a)x.
The plotted curve is generated from the standard-form equation. Hyperbola asymptotes are not drawn — only the two branches are shown. The parabola range is scaled automatically so the curve fits the viewport.
Frequently asked questions
A conic section is any curve produced by the intersection of a plane and a right circular double cone. Algebraically, it is any second-degree (degree-2) polynomial equation in x and y: Ax² + Bxy + Cy² + Dx + Ey + F = 0. The discriminant B² − 4AC determines the type: negative → ellipse/circle, zero → parabola, positive → hyperbola.
Eccentricity e measures how "stretched" a conic is relative to a circle. e = 0: perfect circle. 0 < e < 1: ellipse (more elongated as e → 1). e = 1: parabola (open, unbounded). e > 1: hyperbola (two open branches). All real orbits of solar-system bodies are conics — most planets have e slightly above 0; comets often have e near 1 or above.
Every conic can be defined as the locus of points P where the ratio of the distance from P to a fixed point (the focus) to the distance from P to a fixed line (the directrix) equals the eccentricity e. This single definition unifies all four types: e < 1 gives an ellipse, e = 1 a parabola, e > 1 a hyperbola.
TG we-Calculate Editorial Team. (2026). Conic Sections Calculator — Circle, Ellipse, Parabola, Hyperbola [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/conic-sections-calculator
TG we-Calculate Editorial Team. "Conic Sections Calculator — Circle, Ellipse, Parabola, Hyperbola." TG we-Calculate. 2026. https://we-calculate.com/calculator/conic-sections-calculator.
TG we-Calculate Editorial Team, "Conic Sections Calculator — Circle, Ellipse, Parabola, Hyperbola," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/conic-sections-calculator
@misc{wecalculate_conic_sections_calculator, title = {Conic Sections Calculator — Circle, Ellipse, Parabola, Hyperbola}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/conic-sections-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?