Dodecahedrons are one of the five Platonic solids. These figures are regular, so all their faces have the same shape and all their sides have the same dimensions. We can calculate the volume of dodecahedrons using a standard formula.
In this article, we will learn how to calculate the volume of a dodecahedron. We will learn how to derive the volume formula and use it to solve several practice problems.
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Learning to calculate the volume of a dodecahedron with examples.
Formula for the volume of a dodecahedron
A dodecahedron is a regular three-dimensional figure, so all of its faces have the same shape and all of its sides have the same length. Therefore, we can calculate its volume using the following formula:
| $$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$ |
where a is the length of one of the sides of the dodecahedron.
We can also simplify this formula by getting an approximation of the fraction on the right-hand side of the formula. Thus, we can write:
$latex V\approx 7.663{{a}^3}$
Volume of a dodecahedron – Examples with answers
The following examples are solved by applying the formula for the volume of a dodecahedron. Try to solve the problems yourself before looking at the solution.
EXAMPLE 1
What is the volume of a dodecahedron that has sides with a length of 2 inches?
Solution
We can use the formula for the volume of the dodecahedron with the value a=2. Therefore, we have:
$$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$V=\frac{15+7\sqrt{5}}{4}{{2}^3}$$
$$V=\frac{15+7\sqrt{5}}{4}8$$
$latex V=7.663\times 8$
$latex V=61.3$
The volume of the given dodecahedron is $latex 61.3~{{in}^3}$.
EXAMPLE 2
If a dodecahedron has sides 3 inches long, what is its volume?
Solution
Applying the volume formula with the value a=3, we have:
$$ V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$ V=\frac{15+7\sqrt{5}}{4}{{3}^3}$$
$$ V=\frac{15+7\sqrt{5}}{4}27$$
$latex V=7.663\times 27$
$latex V=206.9$
The volume of the dodecahedron is $latex 206.9~{{in}^3}$.
EXAMPLE 3
Find the volume of a dodecahedron that has sides with a length of 8 ft.
Solution
Using the value of a=8 in the volume formula, we have:
$$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$V=\frac{15+7\sqrt{5}}{4}{{8}^3}$$
$$V=\frac{15+7\sqrt{5}}{4}512$$
$latex V=7.663\times 512$
$latex V=3923.5$
The volume of the given dodecahedron is $latex 3923.5~{{ft}^3}$.
EXAMPLE 4
If the volume of a dodecahedron is equal to $latex 698.3~{{in}^3}$, what is the length of one of its sides?
Solution
In this case, we have the volume and we want to calculate the length of one of its sides. Thus, we use the volume formula and solve for a:
$$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$698.3=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$latex 698.3=7.663{{a}^3}$
$latex 91.13={{a}^3}$
$latex a=4.5$
The length of one of the sides of the dodecahedron is 4.5 in.
EXAMPLE 5
Find the length of the sides of a dodecahedron that has a volume of $latex 1077.5~{{ft}^3}$.
Solution
We use the volume formula and solve for a:
$$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$1077.5=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$latex 1077.5=7.663{{a}^3}$
$latex 140.61={{a}^3}$
$latex a=5.2$
The length of one of the sides of the dodecahedron is 5.2 ft.
EXAMPLE 6
If a dodecahedron has a volume of $latex 2517.4 {{in}^3}$, what is the length of one of its sides?
Solution
Applying the volume formula, we can use V=2517.4 and solve for a:
$$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$2517.4=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$latex 2517.4=7.663{{a}^3}$
$latex 328.51={{a}^3}$
$latex a=6.9$
The length of one of the sides of the dodecahedron is 6.9 in.
EXAMPLE 7
What is the volume of a dodecahedron that has sides with a length of 10 ft?
Solution
We use the formula for the volume of a dodecahedron with the value a=10:
$$V=\frac{15+7\sqrt{5}}{4}{{a}^3}$$
$$V=\frac{15+7\sqrt{5}}{4}{{10}^3}$$
$$V=\frac{15+7\sqrt{5}}{4}1000$$
$latex V=7.663\times 1000$
$latex V=7663$
The volume of the given dodecahedron is $latex 7663~{{ft}^3}$.
Volume of a dodecahedron – Practice problems
Solve the following practice problems by applying what you have learned about the volume of a dodecahedron. In case you need help, you can look at the solved examples above.
See also
Interested in learning more about dodecahedrons? Take a look at these pages:
