Conic Sections Calculator
Select a conic section type, enter your axis lengths or focal parameter, and instantly get eccentricity, linear eccentricity, foci locations, vertices, semi-latus rectum, focal parameter and, for hyperbolas, asymptote slopes. All four classical conic sections are covered: circle, ellipse, parabola and hyperbola in both horizontal and vertical orientations. The Show Your Work panel walks through every formula step by step.
Formula
Worked example
Ellipse with a = 5, b = 3: c = sqrt(25 - 9) = 4, e = 4/5 = 0.8, semi-latus rectum = 9/5 = 1.8, focal parameter = 9/4 = 2.25, foci at (-4, 0) and (4, 0), directrix at x = -6.25 and x = 6.25.
What are conic sections?
Conic sections are the curves formed when a flat plane slices through a double cone at different angles. Depending on the angle, the cross-section is a circle, an ellipse, a parabola or a hyperbola. These four curves appear throughout mathematics, physics and engineering: planetary orbits are ellipses, satellite dish reflectors and telescope mirrors use parabolas, cooling towers are built on hyperbolic cross-sections, and circles are everywhere in geometry. The word "conic" comes from the Greek "konos," meaning cone.
Eccentricity: the single number that defines the shape
Eccentricity (e) is the most important single parameter of any conic. For a circle e = 0, reflecting perfect symmetry. As e approaches 1 from below the ellipse becomes more elongated and the foci move further apart. At e = 1 the curve becomes a parabola - open and unbounded but still having a single axis of symmetry. Above e = 1 the curve is a hyperbola with two separate branches; as e grows the branches open ever wider. Eccentricity is defined as c/a for ellipses and hyperbolas, where c is the distance from center to focus and a is the semi-major or semi-transverse axis. For a parabola the definition is the ratio of distance to focus over distance to directrix, which always equals 1.
Key properties: foci, directrix, latus rectum and focal parameter
Every conic section has one or two foci (singular: focus), which are special points inside or outside the curve. An ellipse is the set of all points whose distances to the two foci sum to a constant (2a). A hyperbola is the set of all points whose distance difference equals 2a. A parabola is equidistant from its single focus and its directrix - a line perpendicular to the axis of symmetry. The semi-latus rectum (l = b^2/a) is the half-length of the chord through the focus perpendicular to the major axis; it appears in the polar form of any conic as r = l/(1 + e cos(theta)). The focal parameter p = b^2/c is the distance from the focus to the directrix.
Practical applications of conic sections
Conic sections are not just abstract curves - they are built into the fabric of the physical world. Johannes Kepler showed in 1609 that planets travel in elliptical orbits with the Sun at one focus. Parabolic mirrors and dishes focus all incoming parallel rays to the focal point, making them ideal for telescopes, radio antennae and solar concentrators. Hyperbolic cooling towers, now common at power plants, are structurally efficient because every horizontal cross-section is a circle, allowing the tower to be assembled from straight beams. The LORAN radio-navigation system located ships by finding the hyperbola on which the difference of distances from two transmitters was constant. Even an arch bridge can follow a parabolic curve for optimal load distribution.
Conic section properties at a glance
| Conic | Standard equation | Eccentricity e | Foci | Directrix |
|---|---|---|---|---|
| Circle | x^2 + y^2 = r^2 | e = 0 | (0, 0) | None |
| Ellipse (h) | x^2/a^2 + y^2/b^2 = 1 (a>b) | 0 < e < 1 | (+/-c, 0) | x = +/-a/e |
| Ellipse (v) | x^2/b^2 + y^2/a^2 = 1 (a>b) | 0 < e < 1 | (0, +/-c) | y = +/-a/e |
| Parabola (up/down) | x^2 = 4py | e = 1 | (0, p) | y = -p |
| Parabola (left/right) | y^2 = 4px | e = 1 | (p, 0) | x = -p |
| Hyperbola (h) | x^2/a^2 - y^2/b^2 = 1 | e > 1 | (+/-c, 0) | x = +/-a/e |
| Hyperbola (v) | y^2/a^2 - x^2/b^2 = 1 | e > 1 | (0, +/-c) | y = +/-a/e |
Key equations and parameter ranges for all four classical conic sections, centred at the origin.
Frequently asked questions
What is the difference between an ellipse and a circle?
A circle is a special case of an ellipse where both axes are equal (a = b), so eccentricity e = 0. In any other ellipse the two axes differ, one focus lies on each side of center, and eccentricity is between 0 and 1. The further e is from 0, the more elongated the ellipse. Every circle is an ellipse, but not every ellipse is a circle.
How do I find the foci of an ellipse?
For an ellipse with semi-major axis a and semi-minor axis b, compute c = sqrt(a^2 - b^2). The foci are at (+/-c, 0) if the major axis is horizontal, or at (0, +/-c) if it is vertical. Add the center coordinates to shift to a non-origin center.
What does eccentricity tell me about a conic section?
Eccentricity measures how far the curve deviates from a perfect circle. e = 0 is a circle; 0 < e < 1 is an ellipse (closer to 0 means more circular, closer to 1 means more elongated); e = 1 is a parabola; e > 1 is a hyperbola (larger e means the branches open more widely). Eccentricity is defined as c/a for ellipses and hyperbolas, where c is the focus-to-center distance.
What are the asymptotes of a hyperbola?
Asymptotes are straight lines that the branches of a hyperbola approach but never touch as they extend to infinity. For a horizontal hyperbola x^2/a^2 - y^2/b^2 = 1 centred at the origin, the asymptotes are y = +/-(b/a)x. For a vertical hyperbola y^2/a^2 - x^2/b^2 = 1, they are y = +/-(a/b)x. The rectangle formed by the semi-axes a and b is used to draw the asymptotes as its diagonals.
What is the semi-latus rectum and why does it matter?
The semi-latus rectum (l) is the half-length of the chord that passes through a focus and is perpendicular to the major axis. It equals b^2/a for an ellipse or hyperbola and 2p for a parabola. It matters because the polar equation of any conic is r = l/(1 + e cos(theta)), where l and e together fully describe the shape. In orbital mechanics, the semi-latus rectum directly relates to a satellite's specific angular momentum.
How is a parabola different from an ellipse?
An ellipse is a closed curve with two foci and eccentricity less than 1. A parabola is an open curve with exactly one focus and eccentricity exactly equal to 1. Every point on a parabola is equidistant from the focus and the directrix; for an ellipse, every point has a constant sum of distances to the two foci. A parabola extends infinitely, while an ellipse is bounded.
Can I enter a general conic equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0?
This calculator focuses on standard-form inputs (semi-axes a, b, and focal parameter p) for all four conic types. To handle the full general form with a cross term (Bxy), you first rotate the coordinate system to eliminate the xy term, then complete the square. For the general form without a cross term, complete the square in x and y to convert to standard form, and you can then enter the resulting a, b values here.