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3D lab · Conic sections

Conic Sections from a Cone in 3D

Slice a double cone with a plane in 3D and watch the circle, ellipse, parabola and hyperbola appear, with the eccentricity e = cos β / cos α.

Last updated: 26 October 2026

Direct answer

Slice a double cone with a plane in 3D and watch the circle, ellipse, parabola and hyperbola appear, with the eccentricity e = cos β / cos α.

Tilt the cutting plane: steeper than the cone's side gives an ellipse, parallel to a side gives a parabola, shallower cuts both nappes in a hyperbola.

Simulation

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Conic Sections from a Cone in 3D

Conic sections

Drag to rotate, two fingers or right-drag to pan, scroll or pinch to zoom. The toolbar has views, cutaway, measure and “show the maths”.

What's happening? Tilt the cutting plane: steeper than the cone's side gives an ellipse, parallel to a side gives a parabola, shallower cuts both nappes in a hyperbola.
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Terms used on this page

Eccentricity (e)

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FAQs

Frequently asked questions

Why is a parabola obtained only for one angle?

The plane must be exactly parallel to a generator of the cone, so β = α.

Why does a hyperbola have two branches?

The plane is shallow enough to cut both nappes of the double cone.

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