# Truncated tetrahedron

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Truncated tetrahedron

The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.

Area and volume

The area "A" and the volume "V" of a truncated tetrahedron of edge length "a" are::$A = 7sqrt\left\{3\right\}a^2 approx 12.1243557a^2$:$V = frac\left\{23\right\}\left\{12\right\}sqrt\left\{2\right\}a^3 approx 2.71057599a^3.$

Cartesian coordinates

Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length &radic;8, are all permutations of (±1,±1,±3) with an even number of minus signs:
* (+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
* (−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
* (−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
* (+3,−1,−1), (+1,−3,−1), (+1,−1,−3)

Use in architecture

Giant truncated tetrahedrons were used for the "Man the explorer" and "Man the producer" theme pavilions in Expo 67. They were made of massive girders of steel bolted together in a geometric lattice. The tetrahedrons were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times.

ee also

*octahedron
*tetrahedron

References

* (Section 3-9)

*
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
* [http://www.lifeisastoryproblem.org/explore/net_truncated_tetrahedron.pdf Printable Net of a Truncated Tetrahedron] [http://www.lifeisastoryproblem.org Life is a Story Problem.org]

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