# Volume of a Triangular Pyramid – Formulas and Examples

The volume of a hexagonal prism is the total space occupied by the prism in three-dimensional space. We can calculate the volume of these prisms by multiplying the area of the base by the height of the prism. The area of the base is equal to the area of a hexagon. The difference between a hexagonal prism and other prisms is the shape of its cross-section.

Here, we will learn about the formula that we can use to calculate the volume of hexagonal prisms. Also, we will use this formula to solve some practice problems.

##### GEOMETRY

Relevant for

Learning about the volume of a triangular pyramid.

See examples

##### GEOMETRY

Relevant for

Learning about the volume of a triangular pyramid.

See examples

## Formula to find the volume of a triangular prism

The volume of any pyramid is equal to the area of the base times the height of the pyramid divided by three. Therefore, the following is the formula for the volume of a pyramid:

$latex \text{Volume} = \frac{1}{3}\times \text{Area base}\times \text{Height}$

In turn, we know that the base of a triangular pyramid is a triangle and the area of any triangle is found by multiplying the length of its base by its height and dividing by two. Therefore, we have:

$latex V=\frac{1}{3}(\frac{1}{2}ba)(h)$

where b is the length of the base of the triangle, a is the length of the height of the triangle, and h is the height of the pyramid.

## Volume of a triangular pyramid – Examples with answers

The formula for the volume of triangular pyramids is used to solve the following examples. 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

A pyramid has a height of 5 m and a triangular base with a base of length 4 m and a height of 3 m. What is its volume?

We have the following information:

• Pyramid height, $latex h=5$
• Base of the triangle, $latex b=4$
• Height of the triangle, $latex a=3$

We use these values in the volume formula:

$latex V=\frac{1}{6}bah$

$latex V=\frac{1}{6}(4)(3)(5)$

$latex V=10$

The volume is equal to 10 m³.

### EXAMPLE 2

What is the volume of a pyramid that has a height of 8 m and a triangular base with a base of 5 m and a height of 6 m?

We have the following values:

• Pyramid height, $latex h=8$
• Base of the triangle, $latex b=5$
• Triangle height, $latex a=6$

Using this information in the volume formula, we have:

$latex V=\frac{1}{6}bah$

$latex V=\frac{1}{6}(5)(6)(8)$

$latex V=40$

The volume is equal to 40 m³.

### EXAMPLE 3

A pyramid has a height of 11 m and a triangular base with a base of length 7 m and a height of 8 m. What is its volume?

From the question, we get the following values:

• Pyramid height, $latex h=11$
• Base of the triangle, $latex b=7$
• Triangle height, $latex a=8$

We use the volume formula with these values:

$latex V=\frac{1}{6}bah$

$latex V=\frac{1}{6}(7)(8)(11)$

$latex V=102.7$

The volume is equal to 102.7 m³.

### EXAMPLE 4

What is the volume of a pyramid with a height of 9 m and a triangular base of a base of 7 m and a height of 9 m?

We have the following values:

• Pyramid height, $latex h=9$
• Base of the triangle, $latex b=7$
• Height of the triangle, $latex a=7$

We substitute these values in the volume formula:

$latex V=\frac{1}{6}bah$

$latex V=\frac{1}{6}(7)(9)(9)$

$latex V=94.5$

The volume is equal to 94.5 m³.

## Volume of a triangular pyramid – Practice problems

Practice using the formula for the volume of triangular pyramids and solve the following problems. If you need help with this, you can look at the solved examples above.