What Are Conic Sections?
A conic section is a curve obtained by intersecting a double-napped right circular cone with a plane. Depending on the angle of the cutting plane, you get four distinct types of curves: circle, ellipse, parabola, and hyperbola. Together, these elegant shapes appear everywhere in nature, engineering, and astronomy.
The Four Types of Conic Sections
- Circle — formed when the cutting plane is perpendicular to the cone's axis. All points are equidistant from the center. Eccentricity = 0.
- Ellipse — formed when the plane cuts through at an angle, intersecting only one nappe. The sum of distances from any point to the two foci is constant. Eccentricity between 0 and 1.
- Parabola — formed when the plane is parallel to one of the generating lines of the cone. Every point is equidistant from the focus and the directrix. Eccentricity = 1 exactly.
- Hyperbola — formed when the plane cuts through both nappes. The difference of distances from any point to the two foci is constant. Eccentricity greater than 1.
Geometric Features of Conic Sections
| Conic | Equation (standard form) | Eccentricity | Key property |
|---|---|---|---|
| Circle | (x−h)² + (y−k)² = r² | e = 0 | All points at equal distance r from center |
| Ellipse | (x−h)²/a² + (y−k)²/b² = 1 | 0 < e < 1 | Sum of distances to foci = 2a (constant) |
| Parabola | (x−h)² = 4p(y−k) | e = 1 | Distance to focus = distance to directrix |
| Hyperbola | (x−h)²/a² − (y−k)²/b² = 1 | e > 1 | Difference of distances to foci = 2a (constant) |
How Does the Conic Sections Calculator Work?
Our calculator accepts the parameters that define each conic section and instantly computes all relevant geometric properties:
- Select the type of conic: circle, ellipse, parabola, or hyperbola.
- Choose your unit system: metric (mm, cm, m, km) or imperial (in, ft, yd, mi).
- Enter the parameters: center/vertex coordinates (h, k) and the shape-specific values (radius r, axes a & b, focal parameter p).
- Click Calculate to instantly see the full equation, foci, directrices, eccentricity, area, perimeter, and more.
Conic Sections in Real Life
- Astronomy & Orbits — Planets follow elliptical orbits (Kepler's first law). Comets may travel in parabolic or hyperbolic paths around the Sun.
- Architecture & Engineering — Parabolic arches distribute loads efficiently. Suspension bridge cables hang in a shape close to a parabola. Cooling towers use hyperbolic cross-sections.
- Optics & Telescopes — Parabolic mirrors in reflecting telescopes focus parallel light rays to a single point. Hyperbolic reflectors are used in Cassegrain telescope designs.
- GPS & Navigation — LORAN (Long Range Navigation) uses hyperbolic lines of position to locate ships and aircraft.
- Projectile Motion — Objects launched near Earth's surface follow a parabolic trajectory (ignoring air resistance).
- Acoustics — Elliptical "whispering galleries" reflect sound from one focus to another with perfect clarity.
Key Formulas at a Glance
Circle
- Equation: (x−h)² + (y−k)² = r²
- Area = πr²
- Circumference = 2πr
- Eccentricity = 0
Ellipse (a ≥ b > 0)
- Equation: (x−h)²/a² + (y−k)²/b² = 1
- Focal distance: c = √(a²−b²)
- Eccentricity: e = c/a
- Area = πab
- Perimeter ≈ π(a+b)(1 + 3h/(10+√(4−3h))), h = ((a−b)/(a+b))² (Ramanujan's approximation)
- Semi-latus rectum: ℓ = b²/a
Parabola
- Vertical: (x−h)² = 4p(y−k) — opens up (p > 0) or down (p < 0)
- Horizontal: (y−k)² = 4p(x−h) — opens right (p > 0) or left (p < 0)
- Focus: (h, k+p) for vertical; (h+p, k) for horizontal
- Directrix: y = k−p for vertical; x = h−p for horizontal
- Latus rectum = 4|p|
- Eccentricity = 1
Hyperbola
- Horizontal: (x−h)²/a² − (y−k)²/b² = 1
- Vertical: (y−k)²/a² − (x−h)²/b² = 1
- Focal distance: c = √(a²+b²)
- Eccentricity: e = c/a (e > 1)
- Asymptotes: y−k = ±(b/a)(x−h) for horizontal orientation
- Semi-latus rectum: ℓ = b²/a
FAQs
Is a circle a special case of an ellipse?
Yes! A circle is an ellipse where the two semi-axes are equal (a = b), so both foci coincide at the center and the eccentricity is exactly 0.
What does eccentricity measure?
Eccentricity (e) measures how "stretched" or "elongated" a conic section is. A circle has e = 0 (perfectly round), an ellipse has 0 < e < 1, a parabola has e = 1, and a hyperbola has e > 1.
What is the directrix?
The directrix is a fixed line used in the definition of a conic. For a parabola, every point on the curve is equidistant from the focus and the directrix. For an ellipse or hyperbola, the directrix relates to the eccentricity: the ratio of distance to the focus versus distance to the directrix equals e for every point on the curve.
Can I use imperial units?
Absolutely. Our calculator supports both metric (mm, cm, m, km) and imperial (in, ft, yd, mi) units. Simply switch the unit system toggle and select your preferred unit before calculating.
What is the latus rectum?
The latus rectum is a chord of the conic section that passes through a focus and is perpendicular to the major axis (or axis of symmetry for a parabola). Its length equals 4|p| for a parabola and 2b²/a for an ellipse or hyperbola.