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Activity solid geometry (second) of

An activity to introduce the representation of solids in perspective and make volume calculations. Initially, students will construct a tetrahedron origami. For this I will show him this little film


and will have a sheet with pictures to do so at their leisure. In passing it may consider to demonstrate that the faces are obtained by folding equilateral triangles. Once folded, the student must identify and calculate the length of the side, the position of the center of the face and height using the Pythagorean theorem. It can thus calculate the volume of the tetrahedron it comes to manufacture. Once manufactured the tetrahedra, the students instructed to assemble tetrahedrons in order to obtain a tetrahedron with dimensions doubled. we quickly realized that it useless to paste the tetrahedra face against face, edge or even against edge, and in fact they should be put on top summit, even if the resulting solid contains a large hole. But what is the shape of this hole? It's hard to see ... Two solutions to better realize: magnetic rods. 1) We remake the four tetrahedra on each other, then removes what is too ... 2) We drew the four tetrahedra in perspective. In passing, we remember the rules: two parallel lines in space are represented by two parallel lines on the drawing. On parallel lines, proportions in length are retained. In going back to red edges of the hole, we realize we have shown an octahedron, and its parallel edges are found in FIG. We arrive at the almost magical part of the activity. What is the volume of the octahedron obtained? Students, surely a bit chilled by the calculation of the volume of the tetrahedron, will go against the grain. And yet! Obtained as the tetrahedron is an enlargement of the small initial tetrahedron with a coefficient 2, the volume has increased by 8. However, there are only four small tetrahedrons. So the full volume is equal to 4 times the initial volume. So the empty part is also 4 times the initial volume. The octahedron has a volume four times larger than the tetrahedron same side. In passing, we can see that the tetrahedron can not pave the area, but the tetrahedron and octahedron pave all the space, as illustrated beautifully in this print by Escher, "Planar".