What quantity did you change?
Labs / Mathematics
Slicing the Cone
Every conic section is just one shape cut four ways. Drag to orbit the double cone, then tilt the slicing plane and slide it up and down — the highlighted curve is exactly what the plane carves out. Flip on “Flat conic” to see that same slice laid out flat with its equation and foci.
Labs / Mathematics / Slicing the Cone
Slicing the Cone
Every conic section is just one shape cut four ways. Drag to orbit the double cone, then tilt the slicing plane and slide it up and down — the highlighted curve is exactly what the plane carves out. Flip on “Flat conic” to see that same slice laid out flat with its equation and foci.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
Ellipsee 0.66x²/a² + y²/b² = 1
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Start with a level plane, then tilt it a little. At what moment does the circle stop being a circle — and what happens to its eccentricity?
- Keep tilting. Right when the plane runs parallel to the cone's side the curve snaps open. Why does the ellipse never quite close again after that?
- The parabola sits exactly at eccentricity 1. What does that single number tell you about the balance between the plane's tilt and the cone's slope?
- Tilt past the parabola into the hyperbola. Where does the second branch come from, and what does it have to do with the cone having two halves?
- Slide the plane's height without changing its tilt. The conic grows and shrinks, but does its type ever change? What does that say about which control really decides the shape?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.