Conic Sections Calculator
Calculate properties of circles, ellipses, parabolas, and hyperbolas with their equations and parameters
Calculate Conic Section Properties
A closed curve where the sum of distances from any point to two foci is constant
Half the length of the longest diameter
Half the length of the shortest diameter (must be ≤ a)
Conic Section Results
Standard Equation
Eccentricity Analysis
Example Calculation
Ellipse Example (Earth's Orbit)
Semi-major axis (a): 4 units
Semi-minor axis (b): 2 units
Equation: x²/16 + y²/4 = 1
Calculated Parameters
Eccentricity: e = √(1 - b²/a²) = √(1 - 4/16) = 0.866
Linear eccentricity: c = √(a² - b²) = √(16 - 4) = 3.464
Semi-latus rectum: l = b²/a = 4/4 = 1
Focal parameter: p = b²/(a² - b²) = 4/12 = 0.333
Conic Section Types
Circle
e = 0
All points equidistant from center
Ellipse
0 < e < 1
Sum of distances to foci is constant
Parabola
e = 1
Distance to focus equals distance to directrix
Hyperbola
e > 1
Difference of distances to foci is constant
Key Formulas
Ellipse Eccentricity
e = √(1 - b²/a²)
Hyperbola Eccentricity
e = √(1 + b²/a²)
Linear Eccentricity
c = √|a² ± b²|
Semi-latus Rectum
l = b²/a
Understanding Conic Sections
What are Conic Sections?
Conic sections are curves obtained by intersecting a cone with a plane. Depending on the angle of intersection, different types of curves are formed: circles, ellipses, parabolas, and hyperbolas.
Eccentricity
Eccentricity (e) measures how much a conic section deviates from being circular:
- •e = 0: Circle (perfect symmetry)
- •0 < e < 1: Ellipse (closed curve)
- •e = 1: Parabola (boundary case)
- •e > 1: Hyperbola (open curve)
Key Parameters
Linear Eccentricity (c)
Distance from center to focus
Semi-latus Rectum (l)
Half the length of the chord through focus parallel to directrix
Focal Parameter (p)
Distance from focus to directrix
Applications: Orbital mechanics, telescope mirrors, satellite dishes, architectural arches, and many natural phenomena follow conic section curves.