An octahedron is a three-dimensional figure that can be formed by joining two pyramids at their bases. Octahedra are one of the five Platonic solids. We can calculate the volume of an octahedron by adding the volume of the two pyramids that form it.

In this article, we will look at the formula that we can use to calculate the volume of an octahedron. We will learn how to derive this formula and apply it to solve some problems.

##### GEOMETRY

**Relevant for**…

Learning to calculate the volume of an octahedron with examples.

##### GEOMETRY

**Relevant for**…

Learning to calculate the volume of an octahedron with examples.

## Formula for the volume of an octahedron

Octahedrons are three-dimensional figures made up of eight triangular faces. We can calculate their volume using the following formula:

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

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

### Proof of the formula for the volume of an octahedron

An octahedron can be formed by joining two pyramids at their bases. This means that we can get the volume of an octahedron if we add the volumes of both pyramids.

Since both pyramids are the same, this means we have to find the volume of one pyramid and simply multiply by 2 to get the volume of the octahedron.

Now, to calculate the volume of any pyramid, we can use the following formula:

$latex V_{p}=\frac{A_{b}\times h}{3}$

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

In this case, the base of the pyramid is square, so its area is equal to $latex {{a}^2}$.

The height of the pyramid can be calculated using the Pythagorean theorem and the following diagram:

Since the faces of an octahedron are equilateral triangles, all of its sides have a length of *a*. Therefore, the hypotenuse that we will use is equal to *a*.

Also, since the diagonal of a square is equal to $latex a\sqrt{2}$, half the diagonal, which is equal to one of the legs, is equal to $latex \frac{a\sqrt{2 }}{2}$.

Therefore, the height of the pyramid is:

$latex h=\sqrt{{{a}^2}-{{(\frac{a\sqrt{2}}{2})}^2}}$

$latex h=\sqrt{{{a}^2}-\frac{{{a}^2}}{2}}$

$latex h=\sqrt{\frac{{{a}^2}}{2}}$

$latex h=\frac{a}{\sqrt{2}}$

$latex h=\frac{a \sqrt{2}}{2}$

Multiplying the height by the area of the base and dividing by 3, we get the volume of the pyramid:

$latex V_{p}=\frac{1}{3}\times{{a}^2}\times \frac{a \sqrt{2}}{2}$

$latex V_{p}=\frac{{{a}^3} \sqrt{2}}{6}$

Multiplying the volume of the pyramid by 2, we get the volume of the octahedron:

$latex V=2\times \frac{{{a}^3} \sqrt{2}}{6}$

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

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## Volume of an octahedron – Examples with answers

The formula for the volume of an octahedron is applied to solve the following examples. Try to solve the problems yourself before looking at the solution.

### EXAMPLE 1

If an octahedron has sides 4 m long, what is its volume?

##### Solution

To solve this problem, we have to use the formula for the volume of an octahedron with the length *a*=4. So, we have:

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

$latex V=\frac{{{4}^3} \sqrt{2}}{3}$

$latex V=\frac{64 \sqrt{2}}{3}$

$latex V=30.17$

The volume of the octahedron is $latex 30.17 ~{{m}^3}$.

### EXAMPLE 2

Calculate the volume of an octahedron that has sides with a length of 5 m.

##### Solution

Again, we use the formula for the volume of a tetrahedron. In this case, we use the value *a*=5:

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

$latex V=\frac{{{5}^3} \sqrt{2}}{3}$

$latex V=\frac{125 \sqrt{2}}{3}$

$latex V=58.93$

The volume of the octahedron is $latex 58.93 ~{{m}^3}$.

### EXAMPLE 3

If an octahedron has sides 10 cm long, what is its volume?

##### Solution

We are going to use the formula for the volume of an octahedron using the value *a*=10. Therefore, we have:

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

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

$latex V=\frac{1000 \sqrt{2}}{3}$

$latex V=471.4$

Then, the volume of the given tetrahedron is $latex 471.4 ~{{cm}^3}$.

### EXAMPLE 4

If the volume of an octahedron is equal to $latex 11.5~{{m}^3}$, what is the length of one of its sides?

##### Solution

In this case, we know the volume of the octahedron and we want to calculate the length of the sides. Therefore, we use the volume formula and solve for *a*:

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

$latex 11.5=\frac{{{a}^3} \sqrt{2}}{3}$

$latex 34.5={{a}^3} \sqrt{2}$

$latex 24.4={{a}^3}$

$latex a=2.9$

The length of one of the sides of the octahedron is 2.9 m.

### EXAMPLE 5

Find the length of the sides of an octahedron that has a volume of $latex 22~{{cm}^3}$.

##### Solution

To solve this problem, we have to use the formula for the volume of an octahedron and solve for *a*. Therefore, we have:

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

$latex 22=\frac{{{a}^3} \sqrt{2}}{3}$

$latex 66={{a}^3} \sqrt{2}$

$latex 46.67={{a}^3}$

$latex a=3.6$

The octahedron has sides with a length of 3.6 cm.

### EXAMPLE 6

The volume of an octahedron is $latex 289.5~{{cm}^3}$. What is the length of its sides?

##### Solution

Let’s use the volume formula and solve for *a*. Therefore, we have:

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

$latex 289.5=\frac{{{a}^3} \sqrt{2}}{3}$

$latex 868.5={{a}^3} \sqrt{2}$

$latex 614.12={{a}^3}$

$latex a=8.5$

The octahedron has sides with a length of 8.5 cm.

### EXAMPLE 7

What is the volume of an octahedron that has sides of length 11.7 cm?

##### Solution

Using the volume formula with length *a*=11.7, we have:

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

$latex V=\frac{{{11.7}^3} \sqrt{2}}{3}$

$latex V=\frac{1601.6\sqrt{2}}{3}$

$latex V=755$

The volume of the octahedron is $latex 755~{{cm}^3}$.

## Volume of an octahedron – Practice problems

Solve the following practice problems using the formula for the volume of an octahedron. If you have trouble with these problems, you can look at the examples with answers above.

## See also

Interested in learning more about octahedra? Take a look at theses pages:

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