Tetrahedron

Introduction to Tetrahedron

Let us first understand about Platonic Solids before we discuss Tetrahedron.

Platonic Solids

A platonic solid is a regular convex polyhedron in a three-dimensional space with identical faces consisting of congruent convex regular polygonal faces. The tetrahedron, cube, octahedron, dodecahedron, and icosahedron are the five solids that follow this criterion.


Tetrahedron

In geometry, a tetrahedron is known as a triangular pyramid. It is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. In simple words, a tetrahedron is a Platonic solid which has a three-dimensional shape having all faces as triangles.


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Tetrahedron Characteristics:

  • A Tetrahedron will have four sides (tetrahedron faces), six edges (tetrahedron edges) and 4 corners.

  • All four vertices are equally distant from one another.

  • Three edges intersect at each vertex.

  • It has six symmetry planes.

  • A tetrahedron has no parallel faces, unlike most platonic solids.

  • On all of its sides, a regular tetrahedron has equilateral triangles.

Tetrahedral Structures

1. Right and oblique tetrahedrons

We can define a tetrahedron as either a right tetrahedron or an oblique tetrahedron. If the tetrahedron's apex is immediately above the base's centre, it is the right tetrahedron. If not, it is a tetrahedron that is oblique. The line segment is perpendicular to the base, which is the height of the tetrahedron, from the apex to the middle of the base of the right tetrahedron.


Right Tetrahedron

Oblique Tetrahedron

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Directly above the center of the base is the apex. 

The apex is not directly above the base's center.


2. Regular and Irregular Tetrahedrons

It is also possible to categorise a tetrahedron as regular or irregular. If equilateral triangles are the four faces of a tetrahedron, then the tetrahedron is a regular tetrahedron. It is irregular, otherwise. All the edges of a regular tetrahedron are equal in length and are congruent to each other on all the faces of a tetrahedron. A regular tetrahedron is a proper tetrahedron as well. An irregular tetrahedron is also an oblique tetrahedron.


Regular Tetrahedron

Irregular Tetrahedron

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All faces are equilateral triangles.

Not all faces are equilateral triangles.


Regular Tetrahedron Formulas

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Area of One Face of Tetrahedron:

\[A = \frac{1}{4} \sqrt{3} a^{2}\]

Area of Tetrahedron:

\[A = \sqrt{3} a^{2}\]

Slant Height of Tetrahedron:

\[h = (\frac{\sqrt{3}}{2})a\]

Altitude of a Tetrahedron:

\[h = \frac{a\sqrt{6}}{3}\]

Volume of Tetrahedron:

\[V = \sqrt{a^{3}}{6\sqrt{2}}\]


Make Your Own Tetrahedron:

Follow this procedure to make a tetrahedron on your own. 

  1. First, let’s take a sheet or paper. 

  2. We’ll make similar lines on the sheet mentioned in the paper below.

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  1. Then we’ll cut the sheet in the edges and fold it as guided in the figure shown below to get a tetrahedral shape.  

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  1. The folded paper will form a tetrahedron.

Do You Know?

  • As we know, a Tetrahedron can be prepared, like all convex polyhedra, by folding a single sheet of paper. It is a polyhedron with the fewest number of faces that can be formed. As shown below, Tetrahedron also has two such different networks.

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  • There will be a sphere for all of the Tetrahedrons on which all four vertices will lie, and another sphere will be tangent to the faces of the Tetrahedrons. 

  • The simplest Tetrahedron is constructed of four equal-sided triangles: one is used as the base, and the other three are fitted to it and each other to make a pyramid. But Egypt's great pyramids are not tetrahedrons. They have a square base and four triangular faces instead and are thus five-faced rather than four-faced. 

  • Should we think of Tetrahedron as the Dice? Yes! Yes! There is an equal probability for a tetrahedron that has four equal faces to fall on any face. Actually, from all the Platonic Solids, you may make equal dices.

Solved Problems:

Q1. What is the volume of a Regular tetrahedron, if it's TSA is \[25\sqrt{3}\]?

Ans: First, let’s find out the length of the edge.

So, we know that TSA of Tetrahedron is,

\[A =\sqrt{3}a^{2}\]

By substituting TSA here, we get 

\[25\sqrt{3} = \sqrt{3}a^{2}\]

\[25 = a^{2}\]

Hence, a = 5 cm

Now, Volume \[V = \frac{a^{3}}{6\sqrt{2}}\]

V ≈ 14.67 cm3

FAQs (Frequently Asked Questions)

1. What are the Properties of a Tetrahedron?

Ans: A pyramid with one triangular base and three triangular sides, called lateral faces, is a tetrahedron. A common vertex called the apex is shared by the lateral faces. Typically, there are 4 sides, 6 edges, and 4 vertices of a tetrahedron. It is the polyhedron with the lowest number of faces that can be formed.

2. Is a Tetrahedron a Prism?

Ans: In geometry, a convex uniform 4-polytope is a tetrahedral prism. There are 6 polyhedral cells in this 4-polytope, 2 tetrahedra connected by 4 triangular prisms. It has 14 faces, 8 triangular and 6 circular respectively. It consists of 16 edges and 8 vertices.

3. How Many Tetrahedrons are in a Cube?

Ans: When we triangulate it differently, by cutting off every other vertex, the cube can be divided into just five tetrahedra. The middle Tetrahedron is regular.