A tetrahedron is a regular pyramid. This means that we can calculate its volume by multiplying the area of its base by the height of the tetrahedron and dividing by three.

In this article, we will learn about the formula to find the volume of a tetrahedron. We will learn how to derive this formula and use it to solve some practice problems.

##### GEOMETRY

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Learning to calculate the volume of a tetrahedron with examples.

##### GEOMETRY

**Relevant for**…

Learning to calculate the volume of a tetrahedron with examples.

## Formula for the volume of a tetrahedron

Tetrahedra are three-dimensional figures made up of four triangular faces. Since the tetrahedron is a triangular pyramid, we can calculate its area by multiplying the area of its base by the length of its height and dividing by 3.

The formula for the volume of a regular tetrahedron is:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$ |

### Proof of the formula for the volume of a tetrahedron

As we mentioned earlier, tetrahedra are triangular pyramids. Also, the area of any pyramid can be calculated by multiplying the area of its base by the height of the pyramid and dividing by three. Therefore, we have:

$latex V=\frac{1}{3}A_{b}H$

where, $latex A_{b}$ is the area of the base and *H* is the height of the tetrahedron.

The base of a tetrahedron is an equilateral triangle and we know that the area of any triangle is equal to one-half the base multiplied by the height. Then, we have:

$latex A_{b}=\frac{1}{2}bh$

The base of the triangle is equal to one of the sides of the tetrahedron, a. Also, the height of an equilateral triangle is equal to $latex \frac{\sqrt{3}}{2}a$, where *a* is the length of one of the sides. Therefore, we have

$latex A_{b}=\frac{1}{2}bh$

$latex A_{b}=\frac{1}{2}a\left( \frac{\sqrt{3}}{2}a\right)$

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

Finally, we have that the height of a tetrahedron is equal to:

$latex H=\frac{\sqrt{6}}{3}a$

Substituting all this into the formula for the volume of a tetrahedron, we have:

$latex V=\frac{1}{3}A_{b}H$

$latex V=\frac{1}{3}\left(\frac{1}{2}bh\right)\left(\frac{\sqrt{6}}{3}a\right)$

$latex V=\frac{{{a}^3}\sqrt{18}}{36}$

$latex V=\frac{{{a}^3}\sqrt{9\times 2}}{36}$

$latex V=\frac{3{{a}^3}\sqrt{2}}{36}$

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

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## Volume of a tetrahedron – Examples with answers

The formula for the volume of a tetrahedron is used to solve the following examples. Each example has its respective solution, but try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

If a tetrahedron has sides 3 m long, what is its volume?

##### Solution

To find the volume of the given tetrahedron, we can simply apply the volume formula by substituting *a*=3. Therefore, we have:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex V=\frac{{{3}^3}\sqrt{2}}{12}$

$latex V=\frac{9\sqrt{2}}{12}$

$latex V=1.06$

The volume of the tetrahedron is $latex 1.06 {{m}^3}$.

### EXAMPLE 2

A tetrahedron has sides with a length of 20 cm. Calculate its volume.

##### Solution

We have that *a*=20. So, we use the formula for the volume of a tetrahedron substituting the given length:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex V=\frac{{{20}^3}\sqrt{2}}{12}$

$latex V=\frac{8000\sqrt{2}}{12}$

$latex V=942.8$

The volume of the tetrahedron is $latex 942.8 {{cm}^3}$.

### EXAMPLE 3

What is the volume of a tetrahedron that has sides with a length of 10 m?

##### Solution

We use the volume formula substituting the value of the length of the sides. Therefore, we have:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex V=\frac{{{10}^3}\sqrt{2}}{12}$

$latex V=\frac{1000\sqrt{2}}{12}$

$latex V=117.9$

The volume of the tetrahedron is $latex 117.9 {{m}^3}$.

### EXAMPLE 4

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

##### Solution

In this case, we have the value of the volume and we want to obtain the length of one of its sides. Therefore, we can use the tetrahedron volume formula and solve for *a*:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex 1000=\frac{{{a}^3}\sqrt{2}}{12}$

$latex 12000={{a}^3}\sqrt{2}$

$latex 12000={{a}^3}\sqrt{2}$

$latex 8485.3={{a}^3}$

$latex a=20.4$

So, the sides of the tetrahedron are 20.4 m long.

### EXAMPLE 5

The volume of a tetrahedron is equal to $latex 400 {{m}^3}$. What is the length of its sides?

##### Solution

Similar to the previous problem, we are going to use the formula for the volume of a tetrahedron and then solve for *a*:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex 400=\frac{{{a}^3}\sqrt{2}}{12}$

$latex 4800={{a}^3}\sqrt{2}$

$latex 4800={{a}^3}\sqrt{2}$

$latex 3394.1={{a}^3}$

$latex a=15.03$

Therefore, the sides of the tetrahedron measure 15.03 m.

### EXAMPLE 6

If a tetrahedron has sides with a length of $latex \sqrt[3]{2}$ m, what is its volume?

##### Solution

We are going to use the volume formula with the given length. Therefore, we have:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex V=\frac{{{(\sqrt[3]{2})}^3}\sqrt{2}}{12}$

$latex V=\frac{2\sqrt{2}}{12}$

$latex V=0.236$

The volume of the tetrahedron is $latex 0.236 {{m}^3}$.

### EXAMPLE 7

Find the volume of a tetrahedron that has sides with a length of 9.5 cm.

##### Solution

We use the value of *a*=9.5 in the formula for the volume of a tetrahedron. Therefore, we have:

$latex V=\frac{{{a}^3}\sqrt{2}}{12}$

$latex V=\frac{{{9.5}^3}\sqrt{2}}{12}$

$latex V=\frac{857.375\sqrt{2}}{12}$

$latex V=101.04$

The volume of the tetrahedron is $latex 101.04 {{m}^3}$.

## Volume of a tetrahedron – Practice problems

Use the formula for the volume of a tetrahedron to solve the following practice problems. If you have problems with these exercises, you can study the solved examples above.

## See also

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

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