matematicas visuales visual math

A regular tetrahedron is a polyhedron with four equilateral triangular faces, four vertices and six edges. It is a Platonic solid.

Tetrahedron plane net: tetrahedron | matematicasVisuales
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.

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:

Tetrahedron plane net: plane net of an tetrahedron by Durer | matematicasVisuales

"[Durer] He introduces a technique of conveying information about three-dimensional objects on a flat surface via paper-folding 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 three-dimensional 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:

Tetrahedron plane net: developing tetrahedron | matematicasVisuales
Tetrahedron plane net: developing tetrahedron | matematicasVisuales
Tetrahedron plane net: developing tetrahedron | matematicasVisuales

In the next application we can see the plane development of a tetrahedron as Durer did:

Tetrahedron plane net: developing tetrahedron | matematicasVisuales
Tetrahedron plane net: developing tetrahedron | matematicasVisuales
Tetrahedron plane net: developing tetrahedron | matematicasVisuales
Tetrahedron plane net: developing tetrahedron | matematicasVisuales

You can download the plane net of this tetrahedron and build your own.

Tetrahedron plane net: cardboard tetrahedron | matematicasVisuales

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.

MORE LINKS

Plane developments of geometric bodies: Octahedron
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 .
Volume of an octahedron
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.
Chamfered Cube
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.
Plane developments of geometric bodies (1): Nets of prisms
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.
Plane developments of geometric bodies (2): Prisms cut by an oblique plane
Plane nets of prisms with a regular base with different side number cut by an oblique plane.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
Sections in Howard Eves's tetrahedron
In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its cross-section areas and its volume.
Sections on a tetrahedron
Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these cross-sections.
Sections in the sphere
We want to calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.