🔺 Conic Sections Calculator
Calculate properties of circles, ellipses, parabolas, and hyperbolas
How to Use This Calculator
Select Conic Section Type
Choose the type of conic section: Circle, Ellipse, Parabola, or Hyperbola.
Enter Parameters
Input the required parameters for your selected conic section type (e.g., radius for circle, semi-axes for ellipse).
Click Calculate
Press the "Calculate" button to get properties like area, eccentricity, equation, and other characteristics.
Formulas
Circle
x² + y² = r²
Area = πr², Circumference = 2πr
Ellipse
x²/a² + y²/b² = 1
Area = πab, Eccentricity = √(1 - b²/a²)
Parabola
x² = 4py (opens upward)
Focus: (0, p), Directrix: y = -p
Hyperbola
x²/a² - y²/b² = 1
Eccentricity = √(1 + b²/a²), Foci: (±√(a²+b²), 0)
About Conic Sections Calculator
The Conic Sections Calculator helps you analyze and calculate properties of the four main conic sections: circles, ellipses, parabolas, and hyperbolas. These curves are formed by the intersection of a plane with a cone.
When to Use This Calculator
- Geometry & Mathematics: Study conic sections and their properties
- Engineering: Design parabolic reflectors, elliptical gears, and circular components
- Physics: Analyze orbital mechanics (ellipses) and projectile motion (parabolas)
- Architecture: Design arches, domes, and elliptical structures
- Education: Learn and practice conic section calculations
- Astronomy: Calculate planetary orbits (elliptical paths)
Why Use Our Calculator?
- ✅ All Conic Types: Supports circles, ellipses, parabolas, and hyperbolas
- ✅ Comprehensive Properties: Calculates area, eccentricity, equation, and more
- ✅ Instant Results: Get all properties immediately
- ✅ Educational: Shows standard equations and formulas
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Conic Sections
Conic sections are curves formed by intersecting a plane with a double-napped cone:
- Circle: Plane perpendicular to cone axis (eccentricity = 0)
- Ellipse: Plane cuts cone at an angle (0 < eccentricity < 1)
- Parabola: Plane parallel to one side of cone (eccentricity = 1)
- Hyperbola: Plane cuts both nappes (eccentricity > 1)
- Eccentricity determines the shape: lower = more circular, higher = more elongated
Real-World Applications
Satellites: Planetary orbits are elliptical with the sun at one focus. The eccentricity determines how elliptical the orbit is (0 = circular, close to 1 = very elongated).
Satellite Dishes: Parabolic reflectors focus incoming signals to a single point (the focus), making them ideal for satellite communications.
Whispering Galleries: Elliptical rooms allow sound from one focus to be clearly heard at the other focus, used in architectural acoustics.
Frequently Asked Questions
What are conic sections?
Conic sections are curves formed by the intersection of a plane with a double-napped cone. The four types are: circle, ellipse, parabola, and hyperbola, each with unique mathematical properties.
How do I identify a conic section from its equation?
General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0. If B² - 4AC < 0: ellipse (or circle if A=C, B=0). If = 0: parabola. If > 0: hyperbola.
What is eccentricity?
Eccentricity (e) measures how "stretched" a conic is. Circle: e=0, Ellipse: 0<e<1, Parabola: e=1, Hyperbola: e>1. Higher eccentricity = more elongated shape.
Can a circle be considered an ellipse?
Yes! A circle is a special ellipse where both semi-axes are equal (a = b). It has eccentricity 0, while ellipses have eccentricity between 0 and 1.
What are the foci of an ellipse?
Ellipses have two foci (focus points). The sum of distances from any point on the ellipse to both foci is constant (equal to 2a, the major axis length).
How are parabolas used in real life?
Parabolas are used in satellite dishes (to focus signals), car headlights (reflectors), bridges (suspension cables form parabolic curves), and projectile motion (paths of thrown objects).
What's the difference between a parabola and a hyperbola?
A parabola has one branch and eccentricity = 1. A hyperbola has two branches and eccentricity > 1. Parabolas open in one direction; hyperbolas have two separate curves.