Sections in a Tetrahedron

We have already seen that the volume of a tetrahedron (in general, not regular) we can obtain joining the diagonals of the faces of a parallelepiped is a third the volume of that parallelepiped.

In this applet we can see the sections of a tetrahedron built from a parallelepiped with square base.

In the initial position, the applet displays a regular tetrahedron cut in half by a square section. It is a well-known puzzle formed by these two pieces alike.

It is easy to build these two pieces and make the puzzle (each piece is a compound of two small tetrahedra and half octahedron):

We are considering these cross-sections of a tetrahedron that are rectangles in general. In the middle, the cross-section is a square.

You can calculate the area of those sections.

Each of these tetrahedra are inside of a prism with square base:

If x is the distance between the center and the section:

You can calculate the side a:

And the side b of the section:

Now the area of the section is:

You can do the same calculation in an interesting particular case on the page devoted to the sections of Howard Eves's tetrahedron.

REFERENCES

Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence.

The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
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