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:
| $latex A_{s}=2\sqrt{3}~{{a}^2}$ |
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:
$latex A_{s}=8A_{t}$
where, $latex A_{t}$ 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:
$latex A_{s}=8\times \frac{\sqrt{3}}{4}{{a}^2}$
$latex A_{s}=2\sqrt{3}~{{a}^2}$
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:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex A_{s}=2\sqrt{3}\times {{2}^2}$
$latex A_{s}=2\sqrt{3}\times 4$
$latex A_{s}=13.86$
The surface area of the given octahedron is $latex 13.86~{{m}^2}$.
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:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex A_{s}=2\sqrt{3}\times {{5}^2}$
$latex A_{s}=2\sqrt{3}\times 25$
$latex A_{s}=86.6$
Therefore, the surface area is $latex 86.6~{{cm}^2}$.
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:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex A_{s}=2\sqrt{3}\times {{8}^2}$
$latex A_{s}=2\sqrt{3}\times 64$
$latex A_{s}=221.7$
The surface area of the octahedron is $latex 221.7~{{cm}^2}$.
EXAMPLE 4
If the surface area of an octahedron is $latex 50~{{m}^2}$, 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:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex 50=2\sqrt{3}~{{a}^2}$
$latex 14.43={{a}^2}$
$latex a=3.8$
The sides of the octahedron have a length of 3.8 m.
EXAMPLE 5
An octahedron has a surface area of $latex 73.3~{{m}^2}$. 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:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex 73.3=2\sqrt{3}~{{a}^2}$
$latex 21.16={{a}^2}$
$latex a=4.6$
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 $latex 150~{{cm}^2}$?
Solution
Using the surface area formula, we can solve for a:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex 150=2\sqrt{3}~{{a}^2}$
$latex 43.3={{a}^2}$
$latex a=6.58$
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:
$latex A_{s}=2\sqrt{3}~{{a}^2}$
$latex A_{s}=2\sqrt{3}\times {{13.4}^2}$
$latex A_{s}=2\sqrt{3}\times 179.56$
$latex A_{s}=622$
The surface area of the octahedron is $latex 622~{{m}^2}$.
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:
