What it shows
Slicing a cone with a flat cut gives a circle when the cut is parallel to the base, an ellipse when tilted, a parabola when parallel to the slanting side, and a hyperbola when steeper than the slanting side. Clay cones, each cut one way (two cones joined for the hyperbola), show the four curves side by side.
The science
A cone is the set of lines through a fixed point making a fixed angle with an axis. A plane cutting it meets these lines in a curve whose type depends only on the plane's angle to the axis compared with the cone's half angle. Parallel to the base the cut is a circle; tilt it less than the slant and the curve stays closed, an ellipse; exactly parallel to the slant it never closes, a parabola; steeper than the slant it meets both halves of a double cone, giving the two branches of a hyperbola. All four are the same family of curves, and the ellipse, parabola and hyperbola share one focus and directrix definition, differing only in their eccentricity.
Materials
Cost: under Rs 500. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- About 2 kg of soft modelling clay or air-dry clay, pressed into five paper cone moulds
- A thin wire or a fishing line with handles to cut
- A protractor and card to set the cut angles
- Four display bases
- Poster colours to paint the cut faces
Safety
For this project: An adult must handle the cutting wire and any knife. Keep clay off the floor and out of drains, and wash hands after the work.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
Step by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Make five identical paper cone moulds, open at the wide end, with base radius 5 cm and height 10 cm, so the slant makes about 27 degrees with the axis.
Press clay firmly into the moulds, let it stiffen for an hour until it holds its shape but is still soft enough to cut, and peel off the paper. The fifth cone is the extra one for the hyperbola, so two can be joined tip to tip.
Cut the first cone horizontally half way up with the wire; the face is a circle of radius 2.5 cm.
Cut the second cone at about 15 degrees to the horizontal, passing through both sides; the face is an ellipse. Measure its long and short axes.
Hold a card against the third cone parallel to one slant side and cut along it; the face is a parabola that reaches the base.
Join two cones tip to tip with a dowel and cut vertically, parallel to the axis but off centre; each cone gives one branch of the hyperbola.
Paint each cut face a bright colour, mount the cones with the faces forward, and label each with its name and the cut angle.
Trace each face on paper, mark the focus where you can, and display the equation of each curve in standard form beside it.
What you should see
Four painted faces show a circle, an ellipse, a parabola and the two branches of a hyperbola, with the cut angle labelled so visitors can see the single rule behind all four.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What decides which conic you get?
The angle of the cutting plane to the axis compared with the slant angle of the cone.
02Why does a hyperbola have two branches?
A steep plane cuts both halves of a double cone, one branch on each half.
03Which conic describes a planet's orbit?
An ellipse with the Sun at one focus, by Kepler's first law.
04What is the eccentricity of a circle, a parabola and a hyperbola?
0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola and more than 1 for a hyperbola.
Take it further
Shine a torch at a wall at different angles: the edge of the light patch traces the same four curves.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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