The surface area of a square pyramid is defined as the sum of the areas of all the faces of the pyramid. In these pyramids, we have a square base and four triangular side faces. Given that the surface area is a two-dimensional measure, we use square units to define it. To find a formula for the surface area, we have to find expressions for the areas of all the faces of the pyramid.

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

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

The surface area of a square pyramid is equivalent to the sum of the areas of all the faces of the pyramid. These pyramids have a square face at the base and four lateral triangular faces. Since the square base has sides of the same length, its area is equal to the length of one of the sides squared.

On the other hand, we know that the area of any triangle is equal to one-half the length of the base times the height of the triangle. In square pyramids, the bases of the triangular faces are the sides of the square base.

This means that the bases of the four triangular faces are equal, therefore their areas are equal. Thus, we have the following formula:

$latex A_{S}={{l}^2}+2lh$ |

where *l* represents the length of one of the sides of the square base and *h* represents the slant height of the triangular faces.

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

The following examples are solved using the formula for the surface area of square pyramids. Try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the surface area of a pyramid that has a square base with sides of 3 m and triangular faces with a height of 4 m?

##### Solution

We have the following values:

- Base sides, $latex l=3$
- Height of triangles, $latex h=4$

We use this information in the formula for surface area:

$latex A_{S}={{l}^2}+2lh$

$latex A_{S}={{3}^2}+2(3)(4)$

$latex A_{S}=9+24$

$latex A_{S}=33$

The surface area is 33 m².

**EXAMPLE 2**

If a square pyramid has sides 5 m long and triangular faces 6 m high, what is its surface area?

##### Solution

We have the following information:

- Base sides, $latex l=5$
- Height of triangles, $latex h=6$

We use the formula for the surface area with the given information:

$latex A_{S}={{l}^2}+2lh$

$latex A_{S}={{5}^2}+2(5)(6)$

$latex A_{S}=25+60$

$latex A_{S}=85$

The surface area is 85 m².

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**EXAMPLE 3**

A pyramid has a square base with sides 10 m long and triangular faces 7 m high. What is its surface area?

##### Solution

From the question, we have the following:

- Base sides, $latex l=10$
- Height of triangles, $latex h=7$

We solve using the formula for surface area:

$latex A_{S}={{l}^2}+2lh$

$latex A_{S}={{10}^2}+2(10)(7)$

$latex A_{S}=100+140$

$latex A_{S}=240$

The surface area is 240 m².

**EXAMPLE 4**

What is the surface area of a square pyramid that has sides 11 m long and triangular faces 12 m high?

##### Solution

We have the following information:

- Base sides, $latex l=11$
- Height of triangles, $latex h=12$

We use the formula for surface area with these values:

$latex A_{S}={{l}^2}+2lh$

$latex A_{S}={{11}^2}+2(11)(12)$

$latex A_{S}=121+264$

$latex A_{S}=385$

The surface area is 385 m².

## Surface area of a square pyramid – Practice problems

Use the formula for the surface area of square pyramids to solve 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
- 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
- Surface Area of a Pentagonal Pyramid – Formulas and Examples – Mechamath
- Parts of a Geometric Pyramid – Mechamath

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