Octahedra are one of the five Platonic solids. These geometric figures are regular and are made up of eight congruent triangular faces. Therefore, to calculate their surface area, we have to calculate the area of one face and multiply it by eight.

Here, we will look at the formula for the surface area of an octahedron. We will learn how to prove this formula and use it to solve some practice problems.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area of an octahedron with examples.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area of an octahedron with examples.

## Formula for the surface area of an octahedron

Octahedrons are three-dimensional figures made up of eight triangular faces. The octahedron is one of the five Platonic solids. To calculate the surface area of an octahedron, we use the following formula:

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

### Proof of the formula for the surface area of an octahedron

We can obtain the formula for the surface area of an octahedron by considering that the surface area of any three-dimensional figure is equal to the sum of the areas of all its faces.

In the case of octahedrons, we have eight congruent triangular faces. That is, we have eight faces with the same shape and the same dimensions, so the surface area is:

where, is the area of each triangular face.

Also, when we talk about an octahedron, we usually mean a regular octahedron. If this is the case, each triangular face is an equilateral triangle.

Therefore, remembering that the formula for the area of an equilateral triangle is:

we can substitute that value into the surface area formula:

## Surface area of an octahedron – Examples with answers

The following examples are solved by applying the formula for the surface area of an octahedron. Each example has its respective solution, but try to solve the problems yourself first.

### EXAMPLE 1

What is the surface area of an octahedron that has sides with a length of 2 m?

##### Solution

We can solve this problem by using the formula for the surface area of an octahedron with the value *a*=2. Therefore, we have:

The surface area of the given octahedron is .

### EXAMPLE 2

Find the surface area of an octahedron that has sides with a length of 5 cm.

##### Solution

We use the surface area formula, substituting the value *a*=5. So, we have:

Therefore, the surface area is .

### EXAMPLE 3

If an octahedron has sides with a length of 8 cm, what is its surface area?

##### Solution

We apply the surface area formula with the value *a*=8:

The surface area of the octahedron is .

### EXAMPLE 4

If the surface area of an octahedron is , what is the length of its sides?

##### Solution

In this case, we have the surface area of the octahedron and we want to calculate the length of one of its sides. Therefore, we have to use the surface area formula and solve for *a*:

The sides of the octahedron have a length of 3.8 m.

### EXAMPLE 5

An octahedron has a surface area of . Determine the length of one of its sides.

##### Solution

Let’s use the surface area formula with the given surface area and solve for *a*:

The octahedron has sides with a length of 4.6 m.

### EXAMPLE 6

What is the side length of an octahedron that has a surface area of ?

##### Solution

Using the surface area formula, we can solve for *a*:

Each side of the octahedron has a length of 6.58 cm.

### EXAMPLE 7

Find the surface area of an octahedron that has side lengths of 13.4 m.

##### Solution

We use the surface area formula with *a*=13.4. Therefore, we have:

The surface area of the octahedron is .

## Surface area of an octahedron – Practice problems

Solve the following problems by applying what you have learned about the surface area 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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