The apothem of a hexagonal prism can be defined as the line segment that connects the center of the hexagonal base with one of its sides in a perpendicular way. We can calculate the length of the apothem using the volume or surface area of the prism. This is possible because the area of a regular hexagon can be calculated using the length of the apothem and the length of one of its sides.

Here, we will learn about the formulas that we can use to calculate the length of the apothem of a hexagonal prism. In addition, we will use these formulas to solve some problems.

## How to find the apothem of a hexagonal prism?

The length of the apothem of a hexagonal prism can be calculated from the volume or surface area of the prism. This is possible because these two measurements depend on the area of the hexagonal base and at the same time, the area of the hexagonal base can be calculated using the length of its apothem and the length of one of the sides.

Therefore, we need to know both the length of one of the sides of the hexagonal base, as well as the measure of volume or surface area.

Recall that the volume of a hexagonal prism is equal to the area of the base times the height of the prism. Also, the area of a regular hexagon is equal to 3*al*, where *a* is the length of the apothem and *l* is the length of one of the sides. Therefore, we have:

$latex V=3alh$ |

where *a* is the length of the apothem, *l* is the length of one of the sides and *h* is the length of the height of the prism.

On the other hand, the surface area of these prisms is equal to the sum of the areas of all their faces. We have mentioned that the area of a regular hexagon is equal to 3*al*, so the area of both hexagonal bases is equal to 6*al*.

Also, we have six rectangular side faces that are the same. Each of these faces has an area of *lh*, so the total area of these faces is 6*lh*. This means that the total surface area of the prism is:

$latex A_{s}=6al+6lh$ |

The length of the apothem can be calculated by using either of these two formulas that is relevant to the known information and by solving for *a*.

## Apothem of a hexagonal prism – Examples with answers

The following examples are solved using the volume and surface area formulas to find the length of the apothem of hexagonal prisms. Try to solve the problems yourself before looking at the result.

**Note:** The lengths of the apothems and the lengths of the sides of the hexagon, given in the following exercises, do not correspond to real regular hexagons. The values given were chosen to facilitate the learning of the methods used to obtain the length of the apothem.

**EXAMPLE 1**

A hexagonal prism has a volume of 450 m³. If the height of the prism is 6 m and the base has sides of length 5 m, what is its apothem?

##### Solution

We have the following information:

- Volume, $latex V=450$
- Height, $latex h=6$
- Sides, $latex l=5$

We use the volume formula with this information and solve for *a*:

$latex V=3alh$

$latex 450=3a(5)(6)$

$latex 450=90a$

$latex a=5$

The length of the apothem is 5 m.

**EXAMPLE 2**

What is the apothem of a hexagonal prism that has a volume of 1512 m³, a height of 9 m, and sides of length 8 m?

##### Solution

We have the following values:

- Volume, $latex V=1512$
- Height, $latex h=9$
- Sides, $latex l=8$

We use these values in the volume formula and solve for *a*:

$latex V=3alh$

$latex 1512=3a(8)(9)$

$latex 1512=216a$

$latex a=7$

The length of the apothem is 7 m.

**EXAMPLE 3**

What is the length of the apothem of a hexagonal prism that has a surface area of 264 m², a height of 6 m, and sides of length 4 m?

##### Solution

We have the following values:

- Surface area, $latex A_{s}=264$
- Height, $latex h=6$
- Sides, $latex l=4$

We use the surface area formula with this information and solve for *a*:

$latex A_{s}=6al+6hl$

$latex 264=6a(4)+6(6)(4)$

$latex 264=24a+144$

$latex 24a=264-144$

$latex 24a=120$

$latex a=5$

The length of the apothem is 5 m.

**EXAMPLE 4**

A prism has a surface area of 912 m². If its height is equal to 10 m and the length of the sides of its base is equal to 8, what is the length of its apothem?

##### Solution

We have the following:

- Surface area, $latex A_{s}=912$
- Height, $latex h=10$
- Sides, $latex l=8$

We use these values in the formula for surface area and solve for *a*:

$latex A_{s}=6al+6hl$

$latex 912=6a(8)+6(10)(8)$

$latex 912=48a+480$

$latex 48a=912-480$

$latex 48a=432$

$latex a=9$

The length of the apothem is 9 m.

## Apothem of a hexagonal prism – Practice problems

Practice what you have learned about the apothem of hexagonal prisms and solve the following problems. Select your answer obtained and check it to verify that it is correct.

## See also

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

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