Volume of an Octahedron – Formula and Examples

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
Formula for the volume of octahedron

Relevant for

Learning to calculate the volume of an octahedron with examples.

See examples

GEOMETRY
Formula for the volume of octahedron

Relevant for

Learning to calculate the volume of an octahedron with examples.

See 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:

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:

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

where, 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 {{a}^2}.

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

Pyramid with height to find the volume of octahedron

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 a\sqrt{2}, half the diagonal, which is equal to one of the legs, is equal to \frac{a\sqrt{2 }}{2}.

Therefore, the height of the pyramid is:

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

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

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

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

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:

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

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

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

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

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


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?

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

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

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

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

V=30.17

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

EXAMPLE 2

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

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

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

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

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

V=58.93

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

EXAMPLE 3

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

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

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

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

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

V=471.4

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

EXAMPLE 4

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

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:

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

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

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

24.4={{a}^3}

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 22~{{cm}^3}.

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

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

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

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

46.67={{a}^3}

a=3.6

The octahedron has sides with a length of 3.6 cm.

EXAMPLE 6

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

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

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

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

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

614.12={{a}^3}

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?

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

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

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

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

V=755

The volume of the octahedron is 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.

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

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If an octahedron has sides of length 7 cm, what is its volume?

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What is the length of the sides of an octahedron that has a volume of 279.4 ~{{m}^3}?

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If the volume of an octahedron is equal to 1177.5 {{m}^3}, what is the length of its sides?

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What is the volume of an octahedron that has sides with a length of 12 cm?

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See also

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