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 wellknown 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 crosssections of a tetrahedron that are rectangles in general. In the middle, the crosssection 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.
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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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Howard Eves's tetrahedron is Cavalieri congruent with a given sphere. You can see that corresponding sections have the same area. Then the volumen of the sphere is the same as the volume of the tetrahedron. And we know how to calculate this volumen.
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In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
