Dodecahedrons are one of the five platonic solids. These figures are regular, so all their faces have the same shape and the same dimensions. Therefore, we can calculate its surface area if we multiply the area of one of its faces by 12.

Here, we will look at the formula for the surface area of a dodecahedron. We will learn to derive this formula and use it to solve some practice problems.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area of a dodecahedron with examples.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area of a dodecahedron with examples.

## Formula for the surface area of a dodecahedron

A dodecahedron is characterized by having twelve congruent pentagonal faces. That is, all twelve faces have the same dimensions. This three-dimensional figure is one of the five Platonic solids.

We can calculate the surface area of a dodecahedron using the following formula:

$latex A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$ |

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

We can simplify this formula by approximating the expression on the right-hand side of the formula. Therefore, we can write:

$latex A_{s}\approx 20.65{{a}^2}$

### Derivation of the formula for the surface area of a dodecahedron

To find a formula for the surface area of a dodecahedron, we can consider that dodecahedrons are regular figures that have 12 faces with the same shape and dimensions.

Since all 12 pentagonal faces of the dodecahedron have the same dimensions, we just have to find the area of one of the faces and multiply the result by 12 to get the surface area of the dodecahedron.

Now, we can find the Area of a Pentagon using the following formula:

$$A=\frac{1}{4}\sqrt{25+10\sqrt{5}}~{{a}^2}$$

Therefore, when we multiply this formula by 12, we have:

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

## Surface area of a dodecahedron – Examples with answers

The following examples are solved by applying the formula for the surface area of a dodecahedron. Try to solve the problems yourself before looking at the solution.

### EXAMPLE 1

What is the surface area of a dodecahedron that has sides with a length of 1 in?

##### Solution

To solve this problem, we can apply the formula for the surface area of a dodecahedron with the value *a*=1. Therefore, we have:

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex A_{s}=20.65\times {{1}^2}$

$latex A_{s}=20.65$

The surface area of the dodecahedron is $latex 20.65~{{in}^2}$.

### EXAMPLE 2

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

##### Solution

Using the surface area formula with the value *a*=2, we have:

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex A_{s}=20.65\times {{2}^2}$

$latex A_{s}=20.65\times 4$

$latex A_{s}=82.6$

The surface area of the dodecahedron is $latex 82.6~{{m}^2}$.

### EXAMPLE 3

Find the surface area of a dodecahedron that has sides with a length of 6 ft.

##### Solution

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

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex A_{s}=20.65\times {{6}^2}$

$latex A_{s}=20.65\times 36$

$latex A_{s}=743.4$

The surface area of the dodecahedron is $latex 743.4~{{ft}^2}$.

### EXAMPLE 4

If a dodecahedron has a surface area of $latex 120~{{in}^2}$, what is the length of its sides?

##### Solution

In this problem, we know the surface area, and we need to determine the length of one of the sides of the dodecahedron. Therefore, we use the surface area formula and solve for *a*:

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex 120=20.65~{{a}^2}$

$latex 5.81=a^2$

$latex a=2.41$

The dodecahedron has sides with a length of 2.41 in.

### EXAMPLE 5

What is the length of the sides of a dodecahedron that has a surface area of $latex 350~{{ft}^2}$?

##### Solution

Similar to the previous problem, we are going to use the surface area formula and solve for *a*. Therefore, we have:

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex 350=20.65~{{a}^2}$

$latex 16.95=a^2$

$latex a=4.12$

The dodecahedron has sides with a length of 4.12 ft.

### EXAMPLE 6

Find the length of the sides of a dodecahedron that has a surface area of $latex 1300~{{ft}^2}$.

##### Solution

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

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex 1300=20.65~{{a}^2}$

$latex 62.95=a^2$

$latex a=7.93$

The dodecahedron has sides with a length of 7.93 ft.

### EXAMPLE 7

What is the surface area of a dodecahedron that has sides with a length of 7 in?

##### Solution

We use the value *a*=7 in the surface area formula. Therefore, we have:

$$A_{s}=3\sqrt{25+10\sqrt{5}}~{{a}^2}$$

$latex A_{s}=20.65~{{a}^2}$

$latex A_{s}=20.65\times {{7}^2}$

$latex A_{s}=20.65\times 49$

$latex A_{s}=1011.85$

The surface area of the dodecahedron is $latex 1011.85~{{mm}^2}$.

## Surface area of a dodecahedron – Practice problems

Solve the following practice problems by applying the formula for the surface area of a dodecahedron. Select an answer and click “Check” to check it.

## See also

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

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