# Surface Area of a Hexagonal Pyramid – Formulas and Examples

The surface area of a hexagonal pyramid is defined as the total surface area of the pyramid in three-dimensional space. Surface area is a two-dimensional measure so we use square units to measure it. To calculate the surface area, we have to add the areas of all the faces of the pyramid.

Here, we will learn about the formula that we can use to calculate the surface area of hexagonal pyramids. Then, we will use this formula to solve some practice problems.

##### GEOMETRY

Relevant for

Learning about the surface area of a hexagonal pyramid.

See examples

##### GEOMETRY

Relevant for

Learning about the surface area of a hexagonal pyramid.

See examples

## Formula to find the surface area of a hexagonal pyramid

The surface area is found by adding the areas of all the faces of the three-dimensional figure. These pyramids have a hexagonal face and six lateral triangular faces. Therefore, we have to find expressions for each area of these faces in terms of the dimensions of the pyramid.

The area of the hexagonal base is found using the formula for the area of a hexagon:

$latex A=\frac{3\sqrt{3}}{2}{{l}^2}$

where l represents the length of one of the sides of the hexagon.

On the other hand, the area of triangular faces is found using the formula for the area of any triangle:

$latex A=\frac{1}{2}b\times h$

where b is the length of the triangle’s base and h is the length of the height of the triangle.

In a hexagonal pyramid, the base of the triangular faces is equal to the sides of the hexagonal base. Furthermore, considering that the six triangular faces are equal, we have the following formula for the surface area of hexagonal pyramids:

$latex A_{s}=\frac{3\sqrt{3}}{2}{{l}^2}+6(\frac{1}{2}b\times h)$

$latex A_{s}=\frac{3\sqrt{3}}{2}{{l}^2}+3lh)$

## Surface area of a hexagonal pyramid – Examples with answers

Each of the following examples is solved using the formula for the surface area of hexagonal pyramids. Try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

If a pyramid has a hexagonal base with sides 1 m long and its triangular faces are 3 m high, what is its surface area?

We have the lengths $latex l=1$ and $latex h=3$. Therefore, we use the formula for surface area with these values:

$latex A_{s}=\frac{3\sqrt{3}}{2}{{l}^2}+3lh$

$latex A_{s}=\frac{3\sqrt{3}}{2}{{(1)}^2}+3(1)(3)$

$latex A_{s}=\frac{3\sqrt{3}}{2}(1)+9$

$latex A_{s}=2.6+9$

$latex A_{s}=11.6$

The surface area is equal to 11.6 m².

### EXAMPLE 2

If a pyramid has a hexagonal base with sides 2 m long and triangular faces 5 m high, what is its surface area?

We have the values $latex l=2$ and $latex h=5$. Using these values in the formula for surface area, we have:

$latex A_{s}=\frac{3\sqrt{3}}{2}{{l}^2}+3lh$

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

$latex A_{s}=\frac{3\sqrt{3}}{2}(4)+30$

$latex A_{s}=10.4+30$

$latex A_{s}=40.4$

The surface area is equal to 40.4 m².

### EXAMPLE 3

What is the surface area of a hexagonal pyramid with sides 4 m long and triangular faces 6 m high?

In the question, we have the lengths $latex l=4$ and $latex h=6$. Using these values in the formula, we have:

$latex A_{s}=\frac{3\sqrt{3}}{2}{{l}^2}+3lh$

$latex A_{s}=\frac{3\sqrt{3}}{2}{{(4)}^2}+3(4)(6)$

$latex A_{s}=\frac{3\sqrt{3}}{2}(16)+72$

$latex A_{s}=41.6+72$

$latex A_{s}=113.6$

The surface area is equal to 113.6 m².

### EXAMPLE 4

If a pyramid has a hexagonal base with sides that are 5 m long and its triangular faces are 10 m high, what is its surface area?

We have the lengths $latex l=5$ and $latex h=10$. Therefore, we use the formula for surface area with these values:

$latex A_{s}=\frac{3\sqrt{3}}{2}{{l}^2}+3lh$

$latex A_{s}=\frac{3\sqrt{3}}{2}{{(5)}^2}+3(5)(10)$

$latex A_{s}=\frac{3\sqrt{3}}{2}(25)+150$

$latex A_{s}=65+150$

$latex A_{s}=215$

The surface area is equal to 215 m².

## Surface area of a hexagonal pyramid – Practice problems

Practice using the formula for the surface area of hexagonal pyramids by solving the following problems. If you need help with this, you can look at the solved examples above.

#### If we have a hexagonal pyramid with sides of 5m and triangular faces of 13m, what is its surface area?  