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
Formula for the surface area of a tetrahedron

Relevant for

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

See examples

GEOMETRY
Formula for the surface area of a tetrahedron

Relevant for

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

See 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:

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:

A_{s}=4A_{t}

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

Diagram of a regular tetrahedron with sides

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:

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:

A_{s}=4A_{t}

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

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


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?

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

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

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

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

A_{s}=43.3

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

EXAMPLE 2

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

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

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

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

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

A_{s}=62.35

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

EXAMPLE 3

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

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

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

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

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

A_{s}=249.4

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

EXAMPLE 4

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

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:

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

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

173.2={{a}^2}

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 1000 {{m}^3}, what is the length of its sides?

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

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

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

577.35={{a}^2}

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 \sqrt{30} m?

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

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

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

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

A_{s}=51.96

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

EXAMPLE 7

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

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

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

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

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

A_{s}=315.67

The surface area of the tetrahedron is 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.

If a tetrahedron has sides of 2m, what is its surface area?

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What is the surface area of a tetrahedron that has sides with a length of 9 cm?

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If the surface area of a tetrahedron is equal to 92.3 {{m}^2}, what is the length of its sides?

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The surface area of a tetrahedron is equal to 180.2 {{m}^2}, what is the length of its sides?

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What is the surface area of a tetrahedron that has sides with a length of 17 m?

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