A tetrahedron is a regular pyramid that is made up of four triangular faces. We can calculate the surface area of a tetrahedron by adding the areas of all four faces.

In this article, we will explore the formula for calculating the surface area of a tetrahedron. We will learn how to derive this formula and use it to solve some practice exercises.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area of a tetrahedron with examples.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area of a tetrahedron with examples.

## Formula for the surface area of a tetrahedron

Tetrahedra are three-dimensional figures that can be considered regular triangular pyramids. In addition, tetrahedra are made up of four congruent faces. This means that we can calculate its surface area by adding the areas of all four faces.

The formula for the surface area of a regular tetrahedron is:

$latex A_{s}=\sqrt{3} ~{{a}^2}$ |

### Proof of the formula for the surface area of a tetrahedron

Since the tetrahedra are triangular pyramids, all four of their faces are congruent. This means that all their faces have the same shape and size. Therefore, we can calculate the surface area if we know the area of one of the faces of the tetrahedron.

This means we have:

$latex A_{s}=4A_{t}$

where, $latex A_{s}$ is the surface area of the tetrahedron and $latex A_{t}$ is the area of one of the triangular faces.

Now, we can calculate the area of one of the faces considering that the faces of a tetrahedron are equilateral triangles. Therefore, we use the formula for the area of an equilateral triangle:

where *a* is the length of one of the sides.

Substituting this into the tetrahedron surface area formula, we have:

$latex A_{s}=4A_{t}$

$latex A_{s}=4\frac{\sqrt{3}}{4}~{{a}^2}$

$latex A_{s}=\sqrt{3}~{{a}^2}$

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## Surface area of a tetrahedron – Examples with answers

The following examples are solved by applying the formula for the surface area of a tetrahedron. Each example has its respective solution, but try to solve the exercises yourself before looking at the answer.

### EXAMPLE 1

What is the surface area of a tetrahedron that has sides with a length of 5 m?

##### Solution

We are going to use the surface area formula given above by substituting *a*=5. Then, we have:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex A_{s}=\sqrt{3}~{{5}^2}$

$latex A_{s}=\sqrt{3}~25$

$latex A_{s}=43.3$

The surface area of the tetrahedron is $latex 43.3 {{m}^2}$.

### EXAMPLE 2

If a tetrahedron has sides 6 m long, what is its surface area?

##### Solution

Using *a*=6 in the surface area formula, we have:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex A_{s}=\sqrt{3}~{{6}^2}$

$latex A_{s}=\sqrt{3}~36$

$latex A_{s}=62.35$

The surface area of the tetrahedron is $latex 62.35 {{m}^2}$.

### EXAMPLE 3

What is the surface area of a tetrahedron that has sides with a length of 12 cm?

##### Solution

Using the surface area formula with *a*=12, we have:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex A_{s}=\sqrt{3}~{{12}^2}$

$latex A_{s}=\sqrt{3}~144$

$latex A_{s}=249.4$

The surface area of the tetrahedron is $latex 249.4 {{m}^2}$.

### EXAMPLE 4

If the surface area of a tetrahedron is equal to $latex 300 {{m}^2}$, what is the length of its sides?

##### Solution

In this case, we have to find the length of one of the sides of the tetrahedron. Then, we can use the surface area formula and solve for *a*:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex 300=\sqrt{3}~{{a}^2}$

$latex 173.2={{a}^2}$

$latex a=13.16$

The length of one of the sides of the tetrahedron is 16.16 m.

### EXAMPLE 5

If the surface area of a tetrahedron is equal to $latex 1000 {{m}^3}$, what is the length of its sides?

##### Solution

Again, we use the surface area formula and solve for *a*. Therefore, we have:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex 1000=\sqrt{3}~{{a}^2}$

$latex 577.35={{a}^2}$

$latex a=24.03$

The length of the sides of the tetrahedron is 24.03 m.

### EXAMPLE 6

What is the surface area of a tetrahedron that has sides with a length of $latex \sqrt{30}$ m?

##### Solution

We use the surface area formula substituting the given value for the length of the sides:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex A_{s}=\sqrt{3}~{{(\sqrt{30})}^2}$

$latex A_{s}=\sqrt{3}~30$

$latex A_{s}=51.96$

The surface area of the tetrahedron is $latex 51.96 {{m}^2}$.

### EXAMPLE 7

Find the surface area of a tetrahedron that has side lengths of 13.5 cm.

##### Solution

Using the value of *a*=13.5 in the surface area formula, we have:

$latex A_{s}=\sqrt{3}~{{a}^2}$

$latex A_{s}=\sqrt{3}~{{13.5}^2}$

$latex A_{s}=\sqrt{3}~182.25$

$latex A_{s}=315.67$

The surface area of the tetrahedron is $latex 315.67 {{m}^2}$.

## Surface area of a tetrahedron – Practice problems

Solve the following problems by applying the formula for the surface area of a tetrahedron. If you have problems with these exercises, you can look at the solved examples indicated above.

## See also

Interested in learning more about tetrahedra? Look at these pages:

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