Plane net of a tetrahedron
A regular tetrahedron is a polyhedron with four equilateral triangular faces, four vertices and six edges. It is a Platonic solid. It was Durer the first to publish plane nets of polyhedra. In his book 'Underweysung der Messung' ('Four Books of Measurement', published in 1525) the author draw plane developments of several Platonic and Archimedean solids, for example, this regular tetrahedron: "[Durer] He introduces a technique of conveying information about threedimensional objects on a flat surface via paperfolding which in modern times is called a net. The method involves developing the surface of a polyhedron onto a plane sheet of paper so that the resulting figure can be cut out as a single connected piece then folded up to form a threedimensional model of the original polyhedron". (Cromwell, p.127)
Playing with the application you can see how a tetrahedron develops into a plane net. This is a very typical plane net of a tetrahedron but it is different from the plane net that Durer drew: In the next application we can see the plane development of a tetrahedron as Durer did: REFERENCES
Erwin Panofsky  The Life and Art of Albrecht Dürer  Princeton University Press
Dan Pedoe  Geometry and the Liberal Arts  St. Martin's Press (p. 76)
Hugo Steinhaus  Mathematical Snapshots  Oxford University Press  Third Edition (p. 197)
Magnus Wenninger  'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell  'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961 (p. 87).
W.W. Rouse Ball and H.S.M. Coxeter  'Matematical Recreations & Essays', The MacMillan Company, 1947.
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The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
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.
In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its crosssection areas and its volume.
Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these crosssections.
