Conic Sections
Calculate eccentricity, linear eccentricity, semi-latus rectum, and focal parameter for circles, ellipses, parabolas, and hyperbolas. Free online conic section geometry calculator with instant results for students and professionals.
About This Calculator
The Conic Sections Calculator computes the key parameters of any conic section -- circles, ellipses, parabolas, and hyperbolas -- based on their semi-axes values. Conic sections are the curves formed by intersecting a double cone with a plane at various angles, creating a family of curves with distinct geometric properties.
This calculator outputs four essential parameters for each conic type: eccentricity (how much the shape deviates from circularity), linear eccentricity (distance from center to focus), semi-latus rectum (chord length through focus), and focal parameter (distance from directrix to focus). For circles, the eccentricity is always 0; for parabolas, it is exactly 1; for ellipses it lies between 0 and 1; for hyperbolas it exceeds 1.
Real-World Applications
Conic sections are fundamental to astronomy (Kepler's laws of planetary motion describe elliptical orbits), optics (parabolic mirrors focus light and radio waves), architecture (hyperboloid cooling towers, elliptical arches), engineering (gear design, satellite dish construction), and physics (projectile motion follows parabolic trajectories). Understanding conic parameters is essential for students of geometry, calculus, physics, and engineering.
Frequently Asked Questions
What are conic sections?
Conic sections are curves obtained by intersecting a double cone with a plane. The four types are circles (plane parallel to base), ellipses (angled cut), parabolas (plane parallel to cone side), and hyperbolas (steep angled cut). Each has unique geometric properties and equations.
What is eccentricity in conic sections?
Eccentricity (e) measures how much a conic section deviates from being circular. Circles have e = 0, ellipses have 0 < e < 1, parabolas have e = 1, and hyperbolas have e > 1. Higher eccentricity means a more stretched or elongated shape.
How do you calculate eccentricity of an ellipse?
For an ellipse with semi-major axis a and semi-minor axis b, eccentricity e = sqrt(1 - b^2/a^2). As the ellipse becomes more circular, e approaches 0. As it becomes more elongated, e approaches 1.
What is the difference between linear eccentricity and focal parameter?
Linear eccentricity (c) is the distance from the center to a focus. The focal parameter (p) is the distance from the directrix to the corresponding focus. For a circle, c = 0 and p = ∞. For a parabola, c is undefined (no center), while p = 2a.
What is the latus rectum of a conic section?
The latus rectum is a chord through a focus parallel to the directrix. Its semi-length (semi-latus rectum, l) is a key parameter: for an ellipse l = b^2/a, for a parabola l = 2a, and for a hyperbola l = b^2/a.
How are conic sections used in real life?
Conic sections appear everywhere: planetary orbits are ellipses (Kepler's first law), satellite dishes and headlights use parabolic reflectors, cooling towers use hyperboloid shapes, and circles are used in wheels, gears, and arches. They are fundamental in astronomy, optics, engineering, and architecture.
What is the equation of a conic section?
The general equation is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. Standard forms: circle x^2 + y^2 = r^2, ellipse x^2/a^2 + y^2/b^2 = 1, parabola y^2 = 4ax, hyperbola x^2/a^2 - y^2/b^2 = 1. The discriminant B^2 - 4AC determines the type.