The volume of a hexagonal prism is the total space occupied by the prism in three-dimensional space. We can calculate the volume of these prisms by multiplying the area of the base by the height of the prism. The area of the base is equal to the area of a hexagon. The difference between a hexagonal prism and other prisms is the shape of its cross-section.
Here, we will learn about the formula that we can use to calculate the volume of hexagonal prisms. Also, we will use this formula to solve some practice problems.
Formula to find the volume of a hexagonal prism
The volume of a hexagonal prism is calculated by multiplying the area of the base by the height of the prism. These polyhedra have hexagonal bases, so we have to calculate the area of a hexagon to find the volume. Recall that the following is the formula for the area of a hexagon:
$latex A=\frac{3\sqrt{3}}{2}{{a}^2}$
where a is the length of one of the sides of the hexagon.
This formula can be derived by dividing the hexagon into six congruent triangles and finding the area of one of the triangles.
Since we have a formula for the area of the base of the rectangular prism, the formula for its volume is:
| $latex V=\frac{3\sqrt{3}}{2}{{a}^2}h$ |
where a is the length of one of the sides of the hexagonal base and h is the length of the height of the prism.
Volume of a hexagonal prism – Examples with answers
The following examples are solved using the formula for the volume of hexagonal prisms. Each example has its respective solution, where the process and reasoning used are detailed.
EXAMPLE 1
What is the volume of a hexagonal prism that has sides of length 4 m and height 6 m?
Solution
We have the following information:
- Sides of the hexagon, $latex a=4$
- Height, $latex h=6$
Using the volume formula, we have:
$latex V=\frac{3\sqrt{3}}{2}{{a}^2}h$
$latex V=\frac{3\sqrt{3}}{2}{{(4)}^2}(6)$
$latex V=\frac{3\sqrt{3}}{2}(16)(6)$
$latex V=249.4$
The volume is 249.4 m³.
EXAMPLE 2
A hexagonal prism has sides of length 5 m and a height of 5 m. What is its volume?
Solution
From the question, we have the following values:
- Sides of the hexagon, $latex a=5$
- Height, $latex h=5$
Substituting these values in the volume formula, we have:
$latex V=\frac{3\sqrt{3}}{2}{{a}^2}h$
$latex V=\frac{3\sqrt{3}}{2}{{(5)}^2}(5)$
$latex V=\frac{3\sqrt{3}}{2}(16)(6)$
$latex V=324.8$
The volume is 324.8 m³.
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EXAMPLE 3
What is the volume of a hexagonal prism that has sides of length 7 m and a height of 8 m?
Solution
We have the following information:
- Sides of the hexagon, $latex a=7$
- Height, $latex h=8$
We use this information in the volume formula:
$latex V=\frac{3\sqrt{3}}{2}{{a}^2}h$
$latex V=\frac{3\sqrt{3}}{2}{{(7)}^2}(8)$
$latex V=\frac{3\sqrt{3}}{2}(49)(8)$
$latex V=1018.4$
The volume is 1018.4 m³.
EXAMPLE 4
A prism has a hexagonal base with sides that are 8 m long and 9 m in height. What is its volume?
Solution
We have the following values:
- Sides of the hexagon, $latex a=8$
- Height, $latex h=9$
Using the volume formula, we have:
$latex V=\frac{3\sqrt{3}}{2}{{a}^2}h$
$latex V=\frac{3\sqrt{3}}{2}{{(8)}^2}(9)$
$latex V=\frac{3\sqrt{3}}{2}(64)(9)$
$latex V=1496.5$
The volume is 1496.5 m³.
Volume of a hexagonal prism – Practice problems
Practice using the formula for the volume of hexagonal prisms by solving the following problems. If you need help with this, you can look at the solved examples above.
See also
Interested in learning more about hexagonal prisms? Take a look at these pages:
