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Truncated tetrahedron

Truncated tetrahedron

This polyhedron is obtained by successively cutting off each of the vertices of the tetrahedron.

A truncated tetrahedron is a semi-regular convex polyhedron with two properties:
1. All faces are regular polygons of two types, a hexagon and a triangle;
2. For any pair of vertices, there is a symmetry of the polyhedron (that is, a motion that translates the polyhedron into itself) that transforms one vertex to another.


The truncated tetrahedron is one of the 13 solids of Archimedes.

Archimedean solids are semiregular polyhedra in the sense that their faces are regular non-similar polygons while the condition of one of the types of spatial symmetry (tetrahedral, octahedral or icosahedral) is preserved.


Mathematical Characteristics of the Truncated Tetrahedron

Truncated tetrahedron sphereA truncated tetrahedron can be placed in a sphere (inscribed), so that each of its vertices touches the inner wall of the sphere

The radius of the described sphere of the Truncated tetrahedron

Truncated tetrahedron R

    where a is the length of the side.

It is not possible to place the sphere inside the Truncated tetrahedron so that it touches all sides simultaneously.
To fulfill this condition, the sphere must touch all squares and triangles.
The distances from the center of the sphere to the middle of the triangles and to the middle of the hexagons are different.

Truncated tetrahedron s

The surface area of ​​the Truncated tetrahedron.

The surface area can be determined by the following formula:


Truncated tetrahedron Sq

Truncated tetrahedron v

The volume of the Truncated tetrahedron is determined by the following formula:

formula Truncated tetrahedron V

how to split a truncated tetrahedron

A truncated tetrahedron can be represented as a group of regular polyhedra. The figure shows that the truncated tetrahedron can be divided into four octahedra and six tetrahedrons.

Accordingly, it is possible to carry out the opposite operation - in the presence of four octahedra and six tetrahedrons, assemble a model of a truncated tetrahedron from parts.

Truncated tetrahedron shape nets

The figure shows the shape net of a truncated tetrahedron:

Truncated tetrahedron ris2

To build a model, you can download the pattern in pdf format and print it on an A4 sheet.
If you intend to print it on a color printer, download the color net.

If you intend to use colored paper, use this net.

In addition, there are other options for coloring the polyhedron, when each of the neighboring faces is painted in its own color. Or a certain number of coloring colors is used, and the same colors do not border each other.

We present to your attention two options for coloring the truncated tetrahedron.


The first option is five-color painting. The faces formed by the tetrahedron have different colors, and the faces formed by cutting off the corners of the tetrahedron are painted in the same color:

download the truncated tetrahedron shape net

Truncated tetrahedron 3

The second option is four-color painting. Opposite faces have the same color

download the truncated tetrahedron shape net

Truncated tetrahedron 4

Video. The process of converting a tetrahedron to a truncated tetrahedron

Truncating all four vertices of the tetrahedron leads to the formation of a truncated tetrahedron. The triangular faces of the original polyhedron lose their area and are converted to hexagonal faces.

Video. Build a polyhedron from a net

This video from our partners, the ART KOSEKOMA team, clearly demonstrates how the pattern is converted into a geometric shape:

Truncated tetrahedron from the set "Magic Edges"

archimedean polyhedra magic edges 18
To assemble polyhedra, we can offer you ready-made patterns that are cut and folded.
To do this, you need to use the details of the set Magic Edges 18.
In addition, in the release itself you will find information about the structure of the polyhedron.
Polyhedra, from the set Magic Edges 18:
Assembling a truncated tetrahedron from the set "Magic Edges"
The rotation of the finished polyhedron assembled from these parts:
Detailed assembly from Alexei Zhigulev (youtube channel - Origami)


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