An icosahedron is one of the five Platonic solids. Icosahedrons are regular figures, so all their faces have the same dimensions. **This means that the surface area of an icosahedron can be calculated by finding the area of one face and multiplying it by 20.**

Here, we will learn about a standard formula that we can use to find the surface area of an icosahedron. Also, we will solve some practice problems where we will use this formula.

##### GEOMETRY

**Relevant for**…

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

##### GEOMETRY

**Relevant for**…

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

## Formula for the surface area of an icosahedron

Dodecahedrons are three-dimensional figures that have twenty congruent triangular faces. This means that all of its faces have the same shape and the same dimensions.

We can use the following formula to determine the surface area of an icosahedron:

$latex A_{s}=5\sqrt{3}~{{a}^2}$ |

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

This formula can be simplified as follows:

$latex A_{s}\approx 8.66{{a}^2}$

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

We can prove the formula for the surface area of an icosahedron by considering that icosahedrons are regular three-dimensional figures that have all of their sides and all of their faces with the same dimensions.

Therefore, we can calculate the surface area of an icosahedron by finding the area of one of the triangular faces and multiplying the result by 20.

The faces are equilateral triangles and the Area of an Equilateral Triangle can be found with the following formula:

$$A=\frac{\sqrt{3}}{4}~a^2$$

Multiplying that formula by 20, we have:

$$A_{s}=20\times \frac{\sqrt{3}}{4}~a^2$$

$$A_{s}=5\sqrt{3}~a^2$$

## Surface area of an icosahedron – Examples with answers

The formula for the surface area of an icosahedron is applied to solve the following examples. Each example has its respective solution.

### EXAMPLE 1

Find the surface area of an icosahedron that has sides with a length of 2 in.

##### Solution

Let’s apply the formula for the surface area of an icosahedron using the length *a*=2:

$latex A_{s}=5\sqrt{3}~a^2$

$latex A_{s}=5\sqrt{3}~(2)^2$

$latex A_{s}=5\sqrt{3}~(4)$

$latex A_{s}=34.64$

The surface area of the icosahedron is $latex 34.64~{{in}^2}$.

### EXAMPLE 2

What is the surface area of an icosahedron that has sides with a length of 5 ft?

##### Solution

Using the surface area formula with length *a*=5, we have:

$latex A_{s}=5\sqrt{3}~a^2$

$latex A_{s}=5\sqrt{3}~(5)^2$

$latex A_{s}=5\sqrt{3}~(25)$

$latex A_{s}=216.51$

The surface area of the given icosahedron is $latex 216.51~{{ft}^2}$.

### EXAMPLE 3

Find the surface area of an icosahedron that has sides with a length of 6 in.

##### Solution

Applying the surface area formula using *a*=6, we have:

$latex A_{s}=5\sqrt{3}~a^2$

$latex A_{s}=5\sqrt{3}~(6)^2$

$latex A_{s}=5\sqrt{3}~(36)$

$latex A_{s}=311.77$

The surface area of the icosahedron is $latex 311.77~{{in}^2}$.

### EXAMPLE 4

If an icosahedron has a surface area of $latex 67.9~{{ft}^2}$, determine the length of one of its sides.

##### Solution

In this case, we already have the surface area of the icosahedron and we need to find the length of one of its sides. Therefore, we can use the surface area formula and solve for *a*:

$latex A_{s}=5\sqrt{3}~a^2$

$latex 67.9=5\sqrt{3}~a^2$

$latex 7.84={{a}^2}$

$latex a=2.8$

Therefore, the sides of the icosahedron have length of *a*=2.8 feet.

### EXAMPLE 5

Find the length of the sides of an icosahedron that has a surface area of $latex 175~{{in}^2}$.

##### Solution

This example is similar to the previous one, so we need to use the given surface area value and solve for *a*:

$latex A_{s}=5\sqrt{3}~a^2$

$latex 175=5\sqrt{3}~a^2$

$latex 20.207={{a}^2}$

$latex a=4.495$

The sides of the icosahedron have a length of 4.495 in.

### EXAMPLE 6

What is the length of the sides of an icosahedron that has a surface area of $latex 461.5~{{ft}^2}$.

##### Solution

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

$latex A_{s}=5\sqrt{3}~a^2$

$latex 461.5=5\sqrt{3}~a^2$

$latex 20.207={{a}^2}$

$latex a=53.289$

The sides of the icosahedron have a length of 53.289 ft.

### EXAMPLE 7

Find the surface area of an icosahedron that has side lengths of 11.7 in.

##### Solution

We substitute the value *a*=11.7 in the surface area formula and we have:

$latex A_{s}=5\sqrt{3}~a^2$

$latex A_{s}=5\sqrt{3}~(11.7)^2$

$latex A_{s}=5\sqrt{3}~(136.89)$

$latex A_{s}=1185.5$

The surface area of the icosahedron is $latex 1185.5~{{in}^2}$.

## Surface area of an icosahedron – Practice problems

Try to solve the following practice problems applying the formula for the surface area of an icosahedron. Click “Verify” to verify that the selected answer is correct.

## See also

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

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