The icosahedron is one of the five platonic solids. This figure is formed by twenty congruent triangular faces. That is, all its faces have the same dimensions. Therefore, we can calculate its volume using the length of one of its sides in a standard formula.

Here, we will learn how to calculate the volume of an icosahedron. We will look at how to obtain its formula, and we will apply it to solve some practice problems.

##### GEOMETRY

**Relevant for**…

Learning to calculate the volume of an icosahedron with examples.

##### GEOMETRY

**Relevant for**…

Learning to calculate the volume of an icosahedron with examples.

## Formula for the volume of an icosahedron

The icosahedron is a regular three-dimensional figure, so all its faces have the same shape and the same dimensions. Therefore, we can calculate the volume of an icosahedron using the length of one of its sides in the following formula:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$ |

where *a* is the length of one of the sides of the icosahedron.

Alternatively, we can simplify this equation by getting an approximation for the fraction on the right-hand side of the equation. Thus, we have:

$latex V=2.1817{{a}^3}$

## Volume of an icosahedron – Examples with answers

The following examples are solved by applying the formula for the volume of an icosahedron. Each example has its answer, but try to solve the problems yourself before looking at the solution.

### EXAMPLE 1

Find the volume of an icosahedron that has sides with a length of 2 ft.

##### Solution

To solve this problem, we are going to use the icosahedron formula with *a*=2. Therefore, we have:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex V=2.1817\times {{2}^3}$

$latex V=2.1817\times 8$

$latex V=17.45$

The volume of the icosahedron is $latex 17.45~{{ft}^3}$.

### EXAMPLE 2

What is the volume of an icosahedron that has sides with a length of 4 ft?

##### Solution

We apply the formula for the volume of an icosahedron using the value *a*=4:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex V=2.1817\times {{4}^3}$

$latex V=2.1817\times 64$

$latex V=139.63$

The volume of the icosahedron is $latex 139.63~{{ft}^3}$.

### EXAMPLE 3

If an icosahedron has sides with a length of 7 in, what is its volume?

##### Solution

Using the formula for the volume of an icosahedron with length *a*=7, we have:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex V=2.1817\times {{7}^3}$

$latex V=2.1817\times 343$

$latex V=748.32$

The volume of the given icosahedron is $latex 748.32~{{in}^3}$.

### EXAMPLE 4

The volume of an icosahedron is $latex 60~{{ft}^3}$. What is the length of one of its sides?

##### Solution

In this example, we know the volume of the icosahedron and we have to find the length of one of its sides. Therefore, we can use the volume formula and solve for *a*:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex 60=2.1817{{a}^3}$

$latex 27.5={{a}^3}$

$latex a=3.02$

The icosahedron has sides with a length of 3.02 ft.

### EXAMPLE 5

Determine the length of the sides of an icosahedron that has a volume of $latex 170~{{in}^3}$.

##### Solution

Let’s use the volume formula and solve for *a*:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex 170=2.1817{{a}^3}$

$latex 77.92={{a}^3}$

$latex a=4.27$

The sides of the icosahedron have a length of 4.27 in.

### EXAMPLE 6

If the volume of an icosahedron is $latex 1117~{{ft}^3}$, what is the length of one of its sides?

##### Solution

Using the volume formula and solving for *a*, we have:

$$V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex 1117=2.1817{{a}^3}$

$latex 512={{a}^3}$

$latex a=8$

The icosahedron has sides with a length of 8 ft.

### EXAMPLE 7

Find the volume of an icosahedron that has sides with a length of 11 ft.

##### Solution

We use the volume formula with length *a*=11. Therefore, we have:

$$ V=\frac{5(3+\sqrt{5})}{12}{{a}^3}$$

$latex V=2.1817{{a}^3}$

$latex V=2.1817\times {{11}^3}$

$latex V=2.1817\times 1331$

$latex V=2903.84$

The volume of the icosahedron is $latex 2903.84~{{ft}^3}$.

## Volume of an icosahedron – Practice problems

Solve the following practice problems by applying the formula for the volume of an icosahedron. See the solved examples above for reference.

## See also

Interested in learning more about icosahedrons? Take a look at these pages:

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