# Tetrahedron

The tetrahedron is a polyhedron with four faces, six edges, and four vertices. It is a three-dimensional figure made up of several polygons which, in this case, are triangles.

The tetrahedron is characterized by being the simplest of polyhedra, and the only one that has less than five sides.

It is worth mentioning that a tetrahedron is a pyramid with a triangular base.

## Elements of a tetrahedron

The elements of a tetrahedron, guiding us from the figure below, are:

- Faces: They are the sides of the tetrahedron that, as we mentioned, are triangles (ABC, ADC, ADB and BDC.
- Edges: It is the union of two faces: AB, AC, AD, BC, CD and DB.
- Vertices: They are those points where the edges meet: A, B, C and D.
- Dihedral angle: It is formed by the union of two faces.
- Polyhedron angle: It is one that is constituted by the sides that coincide in a single vertex.

## Area and volume of the tetrahedron

To know the characteristics of the tetrahedron we can calculate:

- Area: The area of the four triangles that make up the polyhedron would have to be added. In that sense, we must remember that the area of a triangle is calculated by multiplying the base times the height and dividing by 2 (A = bxh / 2)
- Volume: It would be calculated with the following formula

In the formula, b is any face of the polyhedron and h is the height or segment that joins b with its opposite vertex. In addition, the height is perpendicular to the base (they form a right angle or that measures 90º).

## Regular tetrahedron

When all the triangles that make up the tetrahedron are equilateral triangles identical to each other, we are faced with a regular tetrahedron. That is, it would be a case of a regular polyhedron, whose faces are all equal and each one is also a regular polygon.

At this point, we must remember that a regular polygon is one where all the sides have the same length and also their interior angles are also all equal.

Recall then that the area (A) of an equilateral triangle can be calculated using Heron's formula where a, b and c are the measurements of the sides and s is the semiperimeter, which is the perimeter (P) between two.

Then yes:

P = a + b + c = a + a + a = 3a

We have to:

Then, since there are four triangles, we multiply the area of each one by 4 to find the area of the tetrahedron (AT):

On the other hand, if we want to calculate the volume, we must find the height of the polyhedron. To do this, we will be guided by the following image:

First, we will calculate the height (h) of the base (the triangle ABC in this example), which is the segment EB. Angle X measures 90º, so the Pythagorean theorem must be fulfilled, and the hypotenuse (BA), which measures a (the length of all edges in this tetrahedron), is equal to the sum of each leg squared. One of the legs is EA, it is the middle of the segment AC (E cuts the side into two equal parts) and measures a / 2. Also, the second leg is the height of the base (h or EB).

Then, by property of the regular tetrahedron, with F being the center of the triangle, EF will be one third of the segment EB, that is, one third of h.

Next step, to find the height of the tetrahedron (DF), we can apply the Pythagorean theorem again because, as the height is perpendicular, the angle Y is right (it measures 90º).

Looking at the triangle DEF, the hypotenuse is DE, which is the height of the triangle ADC and, since all the faces are equal, it is the same height h of the triangle ABC. In turn, one leg is the height of the tetrahedron (DF), which we will call ht, and the other leg is the segment EF that we have already calculated. Therefore:

Finally, to find the volume of the tetrahedron (V), as we explained previously, we multiply the height of the figure (ht) by the area of the base (A) that is calculated above, and divide it by three:

## Tetrahedron example

Assuming that a tetrahedron is regular and each side of its faces is 20 meters. What is the area (AT) and volume (V) of the figure?

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