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Conic Sections Calculator

Calculate properties of circles, ellipses, parabolas, and hyperbolas. Get equations, eccentricity, foci, area, and interactive graph visualization.

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Horizontal center coordinate
Positive = right/up, Negative = left/down
Enter parameters and press Calculate to see results.
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About

Conic sections arise from slicing a right circular cone with a plane at varying angles. The four resulting curves - circle, ellipse, parabola, and hyperbola - are governed by a single parameter: eccentricity e. When e = 0 the curve is a circle. For 0 < e < 1 it is an ellipse. At e = 1 you get a parabola, and e > 1 produces a hyperbola. Misidentifying the conic type or miscalculating the focal distance c leads to errors in antenna dish design, orbital mechanics, bridge arch geometry, and optical system layout. This calculator computes standard and general form equations, eccentricity, foci, directrices, latus rectum, area, and perimeter using exact closed-form expressions and the Ramanujan perimeter approximation for ellipses.

Limitations: all inputs assume real-valued, non-degenerate conics. Degenerate cases (e.g., a = 0) are rejected. The ellipse perimeter uses Ramanujan's second approximation, accurate to within 0.04% for eccentricities below 0.95. The interactive canvas plots parametric curves at 500 sample points, sufficient for visual accuracy but not CAD-grade precision.

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Formulas

All four conic sections satisfy the polar equation with focus at the origin:

r = l1 − e cos θ

where l is the semi-latus rectum and e is eccentricity.

For an ellipse with semi-major axis a and semi-minor axis b (a ≥ b), the focal distance and eccentricity are:

c = √a2 − b2 ,   e = ca

Ellipse perimeter uses the Ramanujan second approximation:

P ≈ π(a + b) ⋅ (1 + 3h10 + √4 − 3h) ,   h = (a − b)2(a + b)2

For a hyperbola, the relationship inverts to c2 = a2 + b2, with asymptote slopes ±ba.

For a parabola y2 = 4px, the focus is at (p, 0) and the directrix at x = −p.

Variable legend: a = semi-major axis (or transverse semi-axis for hyperbola). b = semi-minor axis (or conjugate semi-axis). c = linear eccentricity (center-to-focus distance). e = eccentricity (dimensionless ratio). p = focal length (vertex-to-focus distance for parabola). r = radius (circle). h, k = center/vertex coordinates. l = semi-latus rectum.

Reference Data

Conic TypeEccentricity eStandard Form (Centered)FociDirectrixArea
Circle0x2 + y2 = r2Center (0, 0)N/Aπr2
Ellipse0 < e < 1x2a2 + y2b2 = 1(±c, 0)x = ±aeπab
Parabola (right)1y2 = 4px(p, 0)x = −pUnbounded
Parabola (up)1x2 = 4py(0, p)y = −pUnbounded
Hyperbola (horiz.)e > 1x2a2 − y2b2 = 1(±c, 0)x = ±aeUnbounded
Hyperbola (vert.)e > 1y2a2 − x2b2 = 1(0, ±c)y = ±aeUnbounded
Key relationships
Ellipsec2 = a2 − b2,   e = ca,   Latus rectum = 2b2a
Hyperbolac2 = a2 + b2,   e = ca,   Asymptotes: y = ±bax
ParabolaFocus-directrix distance = 2p,   Latus rectum = |4p|
CircleSpecial case of ellipse where a = b = r and c = 0

Frequently Asked Questions

Eccentricity e is the ratio of the distance from any point on the curve to a focus versus its distance to the corresponding directrix. A circle has e = 0. An ellipse has 0 < e < 1. A parabola has e = 1 exactly. A hyperbola has e > 1. As e approaches 1 from below, the ellipse becomes increasingly elongated.
Ramanujan's second approximation for ellipse perimeter relies on the parameter h = (a − b)2 / (a + b)2. When b approaches 0 (eccentricity near 1), h approaches 1, and higher-order terms in the exact infinite series become significant. For e < 0.95, relative error stays below 0.04%. Beyond that, use numerical integration (complete elliptic integral of the second kind).
The latus rectum is the chord through a focus perpendicular to the major axis (or axis of symmetry). For an ellipse or hyperbola, its length is 2b2a. For a parabola y2 = 4px, it equals |4p|. The latus rectum controls beam width in parabolic reflectors and orbital semi-parameter in Keplerian mechanics.
Asymptotes define the envelope that the hyperbola approaches at infinity. For a centered horizontal hyperbola, the asymptote slopes are ±ba. In cooling tower design (hyperbolic shells), these slopes govern structural wall angles. In radar and navigation (LORAN), the asymptotic angle determines bearing accuracy at long range. Misidentifying a and b swaps the asymptote orientation.
Yes. A negative p in y2 = 4px opens the parabola to the left instead of the right. The focus moves to (p, 0) which is on the negative x-axis, and the directrix shifts to x = −p (positive x). The sign of p controls orientation. This calculator supports all four orientations: right, left, up, and down.
The general second-degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 requires completing the square for x and y terms separately (when B = 0). The discriminant B2 − 4AC classifies the conic: negative for ellipse/circle, zero for parabola, positive for hyperbola. This calculator provides both standard and expanded general forms for each conic.