The surface area of a pentagonal pyramid represents the entire surface occupied by the pyramid in three-dimensional space. This is a two-dimensional measure, so we use square units to represent it. The surface area of any geometric figure is calculated by adding the areas of all the faces of the figure. In this case, we have one pentagonal face and five triangular faces.

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

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

The surface area is calculated by adding the areas of all the faces of a geometric figure. Pentagonal pyramids have one pentagonal face and five lateral triangular faces. To find the area of the pentagonal face, we use the following formula:

$latex A=1.72{{l}^2}$

where *l* represents the length of one of the sides of the pentagonal base.

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

$latex A=\frac{1}{2}bh$

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

In pentagonal pyramids, the bases of the triangular faces are equal to the length of one of the sides of the base. Also, the five triangular faces are congruent. That means the formula for the surface area of these pyramids is:

$latex A_{s}=1.72{{l}^2}+5(\frac{1}{2}bh)$

$latex A_{s}=1.72{{l}^2}+(\frac{5}{2}lh)$

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

Use the following examples to practice the process used to calculate the surface area of pentagonal pyramids. Each example has its respective solution, but it is recommended that you try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the surface area of a pentagonal pyramid with a height of 5 m and sides of length 1 m?

##### Solution

From the question, we have the lengths $latex h=5$ and $latex l=1$. Using the volume formula with these values, we have:

$latex A_{s}=1.72{{l}^2}+\frac{5}{2}lh$

$latex A_{s}=1.72{{(1)}^2}+\frac{5}{2}(1)(5)$

$latex A_{s}=1.72+12.5$

$latex A_{s}=14.22$

The surface area is equal to 14.22 m².

**EXAMPLE 2**

If a pyramid has a height of 6 m and a pentagonal base with sides of 2 m, what is its surface area?

##### Solution

We look at the lengths $latex h=6$ and $latex l=2$. If we use these values in the volume formula, we have:

$latex A_{s}=1.72{{l}^2}+\frac{5}{2}lh$

$latex A_{s}=1.72{{(2)}^2}+\frac{5}{2}(2)(6)$

$latex A_{s}=6.88+30$

$latex A_{s}=36.88$

The surface area is equal to 36.88 m².

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**EXAMPLE 3**

What is the surface area of a pentagonal pyramid that has sides of 4 m and a height of 10 m?

##### Solution

We use the lengths $latex h=10$ and $latex l = 4$ in the volume formula. Therefore, we have:

$latex A_{s}=1.72{{l}^2}+\frac{5}{2}lh$

$latex A_{s}=1.72{{(4)}^2}+\frac{5}{2}(4)(10)$

$latex A_{s}=27.52+100$

$latex A_{s}=127.52$

The surface area is equal to 127.52 m².

**EXAMPLE 4**

If a pentagonal pyramid has sides that are 6 m long and 12 m high, what is its surface area?

##### Solution

We substitute the lengths $latex h =12$ and $latex l=6$ in the formula for the surface area:

$latex A_{s}=1.72{{l}^2}+\frac{5}{2}lh$

$latex A_{s}=1.72{{(6)}^2}+\frac{5}{2}(6)(12)$

$latex A_{s}=61.92+180$

$latex A_{s}=141.92$

The surface area is equal to 141.92 m².

## Surface area of a pentagonal pyramid – Practice problems

Practice using the pentagonal pyramids surface area formula 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 geometric pyramids? Take a look at these pages:

- Volume of a Square Pyramid – Formulas and Examples – Mechamath
- Surface Area of a Square Pyramid – Formulas and Examples – Mechamath
- Volume of a Hexagonal Pyramid – Formulas and Examples – Mechamath
- Surface Area of a Hexagonal Pyramid – Formulas and Examples – Mechamath
- Volume of a Pentagonal Pyramid – Formulas and Examples – Mechamath
- Parts of a Geometric Pyramid – Mechamath

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