The surface area of a triangular pyramid is equal to the sum of the areas of all the faces of the pyramid. In this type of pyramids, all the faces are triangular, so we have to use the formula for the area of a triangle to get a formula for the surface area.

Here, we will derive a formula for the surface area of triangular pyramids. Also, we will use this formula to solve some problems.

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

We can find the surface area of triangular pyramids by adding the areas of all the faces of the pyramid. A pyramid with a triangular base has a total of four faces, and these four faces are triangular. That means we have to use the formula for the area of a triangle to calculate the areas of the faces.

Recall that the area of any triangle is found by multiplying half the length of its base by the length of its height. Therefore, we need to know the length of the base and the height of each of the triangular faces.

Let’s consider the following diagram:

If the base is an equilateral triangle, which is generally the case, the length of one of these sides is equal to the base of all the faces. Also, if the base is equilateral, the lateral faces have the same area.

Therefore, the area of the base is equal to $latex \frac{1}{2} ba$, where *b* is the length of the base and *a* is the length of the height of the base triangle.

On the other hand, the area of each lateral face is equal to $latex \frac{1}{2} bh$, where *b* is the length of the base and *h* is the slant height of a lateral face. Therefore, the surface area is equal to:

$latex A_{S}=\frac{1}{2}ba+\frac{3}{2}bh$ |

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

The formula for the surface area of a triangular pyramid is used to solve the following examples. Each example has its respective solution, where the process and reasoning used are detailed.

**EXAMPLE 1**

What is the surface area of a pyramid that has a triangular base with sides of length 3 m and a height of 2.6 m and whose lateral faces have a slant height of 4 m?

##### Solution

We have the following lengths:

- Base, $latex b=3$
- Base height, $latex a=2.6$
- Lateral face height, $latex h=4$

Using the formula for the surface area with these lengths, we have:

$latex A_{S}=\frac{1}{2}ba+\frac{3}{2}bh$

$latex A_{S}=\frac{1}{2}(3)(2.6)+\frac{3}{2}(3)(4)$

$latex A_{S}=3.9+18$

$latex A_{S}=21.9$

The surface area is 21.9 m².

**EXAMPLE 2**

A pyramid has a triangular base with sides of 4 m and a height of 3.5 m. If the slant height of its lateral faces is equal to 5 m, what is its surface area?

##### Solution

We have the following values:

- Base, $latex b=4$
- Base height, $latex a=3.5$
- Lateral face height, $latex h=5$

We use these values in the formula for surface area:

$latex A_{S}=\frac{1}{2}ba+\frac{3}{2}bh$

$latex A_{S}=\frac{1}{2}(4)(3.5)+\frac{3}{2}(4)(5)$

$latex A_{S}=7+30$

$latex A_{S}=37$

The surface area is 37 m².

**EXAMPLE 3**

A pyramid has a triangular base with sides of 6 m and a height of 5.2 m. If the slant height of its lateral faces is equal to 6 m, what is its surface area?

##### Solution

From the question, we have the following:

- Base, $latex b=6$
- Base height, $latex a=5.2$
- Lateral face height, $latex h=6$

We use these values in the formula for surface area:

$latex A_{S}=\frac{1}{2}ba+\frac{3}{2}bh$

$latex A_{S}=\frac{1}{2}(6)(5.2)+\frac{3}{2}(6)(6)$

$latex A_{S}=15.6+54$

$latex A_{S}=69.6$

The surface area is 69.6 m².

**EXAMPLE 4**

What is the surface area of a pyramid that has a base with sides of length 7 m and height 6 m and lateral faces of slant height 10 m?

##### Solution

We have the following lengths:

- Base, $latex b=7$
- Base height, $latex a=6$
- Lateral face height, $latex h=10$

Substituting these values in the formula for surface area, we have

$latex A_{S}=\frac{1}{2}ba+\frac{3}{2}bh$

$latex A_{S}=\frac{1}{2}(7)(6)+\frac{3}{2}(6)(10)$

$latex A_{S}=21+90$

$latex A_{S}=111$

The surface area is 111 m².

## Surface area of a triangular pyramid – Practice problems

Practice using the formula for the surface area of triangular pyramids and 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 triangular pyramids? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**