MATHEMATICS2D GeometryMathematics Calculator
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Conic Sections

Calculate properties of circles, ellipses, parabolas, and hyperbolas from their geometric parameters.

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Why: Understanding conic sections helps you make better, data-driven decisions.

How: Enter Conic Type, Unit, Radius (r) to calculate results.

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GEOMETRYConic Sections

Conic Sections โ€” Circle, Ellipse, Parabola, Hyperbola

Calculate properties from direct geometric inputs: radius, semi-axes, or focal length.

Circle โ†’Crescent โ†’Quadrilateral โ†’

๐Ÿ“ Real-World Examples โ€” Click to Load

Inputs

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿ“‹ Key Takeaways

  • โ€ข Conic sections are curves from a plane cutting a double cone: circle, ellipse, parabola, hyperbola
  • โ€ข Circle: all points equidistant from center. Eccentricity e = 0
  • โ€ข Ellipse: sum of distances to two foci is constant. 0 < e < 1
  • โ€ข Parabola: equidistant from focus and directrix. e = 1
  • โ€ข Hyperbola: difference of distances to two foci is constant. e > 1

๐Ÿ’ก Did You Know?

๐Ÿ“กSatellite dishes use parabolic reflectors to focus signals to a single pointSource: Engineering
๐ŸŒPlanetary orbits are ellipses with the Sun at one focus (Kepler's 1st Law)Source: Astronomy
๐ŸŽฏA circle is a special ellipse with a = b (zero eccentricity)Source: Geometry
๐Ÿ”ญCassegrain telescopes use hyperbolic secondary mirrorsSource: Optics
โ›ฒWater fountains follow parabolic arcs due to gravitySource: Physics
๐Ÿš€Escape trajectories are hyperbolic when velocity exceeds escape speedSource: Orbital Mechanics

๐Ÿ“– Formulas Explained

Circle

A=ฯ€r2,C=2ฯ€rA = \pi r^2, \quad C = 2\pi r

Ellipse

A=ฯ€ab,e=ca,c=a2โˆ’b2A = \pi ab, \quad e = \frac{c}{a}, \quad c = \sqrt{a^2 - b^2}

Parabola & Hyperbola

y=x24p(parabola),x2a2โˆ’y2b2=1,โ€…โ€Šc=a2+b2(hyperbola)y = \frac{x^2}{4p} \quad \text{(parabola)}, \quad \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \; c = \sqrt{a^2+b^2} \quad \text{(hyperbola)}

๐ŸŽฏ Expert Tips

Ellipse Axes

a is always the semi-major axis (larger), b the semi-minor. The calculator handles a < b automatically.

Parabola Focus

The focus is p units from the vertex. For satellite dishes, p determines the depth.

Hyperbola Foci

Foci are at ยฑc where cยฒ = aยฒ + bยฒ. Asymptotes have slope ยฑb/a.

Eccentricity

e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola. Measures "flatness".

โš–๏ธ Comparison Table

ConicEccentricityFoci
Circle01 (center)
Ellipse0 < e < 12
Parabola11
Hyperbola> 12

๐Ÿ“Š Quick Facts

4
Conic Types
0โ€“โˆž
Eccentricity
ฯ€
Circle/Ellipse
2
Foci (ellipse/hyperbola)

โ“ FAQ

What is a conic section?

A curve formed by intersecting a plane with a double cone. The angle of the cut determines the type: circle, ellipse, parabola, or hyperbola.

What is eccentricity?

A measure of how much a conic deviates from a circle. e=0 for circle, 0<e<1 for ellipse, e=1 for parabola, e>1 for hyperbola.

How do I find the foci of an ellipse?

Foci are at distance c from center, where cยฒ = aยฒ โˆ’ bยฒ. They lie on the major axis.

What is the parabola focus?

The point p units from the vertex. All rays parallel to the axis reflect through the focus.

Why are hyperbola foci outside the curve?

For a hyperbola, c > a, so foci are beyond the vertices. The curve is defined by |distance to F1 โˆ’ distance to F2| = 2a.

Can a circle have eccentricity?

Yes, e = 0. A circle is an ellipse with a = b, so c = 0.

What is the ellipse perimeter formula?

No closed form exists. We use Ramanujan's approximation: ฯ€(a+b)(1 + 3h/(10+โˆš(4โˆ’3h))) where h = (aโˆ’b)ยฒ/(a+b)ยฒ.

How are conics used in real life?

Satellite dishes (parabola), planetary orbits (ellipse), telescope mirrors (hyperbola), wheels (circle).

โš ๏ธ Disclaimer: Results assume standard-position conics (centered at origin, axes aligned). Real-world applications may use translated/rotated forms.

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