Math resources Geometry Volume

Volume of a triangular prism

# Volume of a triangular prism

Here you will learn about the volume of a triangular prism, including how to calculate the volume and how to find a missing length given the volume.

Students will first learn about the volume of a triangular prism as part of geometry in 7 th grade.

## What is the volume of a triangular prism?

The volume of a triangular prism is how much space there is inside a triangular prism. A triangular prism is a polyhedron (3D shape made from polygons) with two congruent triangular ends connected by three rectangles.

To find this, find the area of the triangular base (cross section) and multiply it by the length.

\text{Volume of a triangular prism} = \text{Area of triangular cross section} \times \text{length}

For example,

Area of triangular cross-section:

\begin{array}{l}\text{Area }=\cfrac{1}{2}bh \\\\ \text{Area }=\cfrac{1}{2} \times 4 \times 5 \\\\ \text{Area }=10\mathrm{~cm}^{2} \end{array}

Volume of triangular prism:

\, \text{ Volume }= \text{Area of triangular cross-section } \times \text{ length}

\begin{array}{l} \text{Volume }=10 \times 11 \\\\ \text{Volume }=110\mathrm{~cm}^{3}\end{array}

Volume is measured in cubic units (for example, \mathrm{mm}^3, \mathrm{~cm}^3, \mathrm{~m}^3 etc).

### Calculating a missing length

Sometimes you might be given the volume and some of the measurements of a triangular prism and need to find the other measurements.

This can be done by substituting the values that are known into the volume of a triangular prism formula and solving the equation that is formed.

For example,

The volume of this triangular prism is 105 \mathrm{~cm}^3. Find the length, x, of the triangular prism.

You will still use the following formula,

\text{Volume of a triangular prism} = \text{area of triangular cross section} \times \text{length}

First, calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula, then solve.

\begin{aligned}& \text { Area of triangle }=\cfrac{1}{2} \times b \times h \\\\ & \text { Area of triangle }=\cfrac{1}{2} \times 2 \times 3 \\\\ & \text { Area of triangle }=3\end{aligned}

\text{Volume of a triangular prism} = \text{area of triangular cross section} \times \text{length}

\begin{aligned}& 105=3 \times x \\\\ & 3 x=105 \\\\ & x=35 \end{aligned}

## Common Core State Standards

How does this relate to 7 th grade math?

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

## How to calculate the volume of a triangular prism

In order to calculate the volume of a triangular prism:

1. Write down the formula.
2. Calculate the area of the triangular cross-section and substitute the values.
3. Solve the equation.
4. Write the answer, including the units.

## Volume of a triangular prism examples

### Example 1: volume of a triangular prism

Find the volume of this triangular prism.

1. Write down the formula.

\text{Volume of a triangular prism} = \text{Area of triangular cross section} \times \text{length}

2Calculate the area of the triangular cross-section and substitute the values.

The base of the triangle is 2 \mathrm{~in} and the height of the triangle is 3 \mathrm{~in}.

\begin{array}{l}\text{Area of triangle }=\cfrac{1}{2} \times b \times h \\\\ \text{Area of triangle }=\cfrac{1}{2} \times 2 \times 3 \\\\ \text{Area of triangle }=3\end{array}

The area of the triangle is 3 \mathrm{~in}^2.

The length of the prism is 7 \mathrm{~in}.

\text{Volume of a triangular prism} = \text{Area of triangular cross section} \times \text{length}

\text{Volume of triangular prism} = 3 \times 7

3Solve the equation.

\begin{aligned}& \text{Volume of triangular prism} =3 \times 7 \\\\ & \text{Volume of triangular prism} = 21 \end{aligned}

4Write the answer, including the units.

The measurements on this triangular prism are in inches, so the volume will be measured in cubic inches.

\text { Volume }=21 \mathrm{~in}^3

### Example 2: triangular prism with isosceles triangle

Find the volume of the triangular prism.

Write down the formula.

Calculate the area of the triangular cross-section and substitute the values.

Solve the equation.

Write the answer, including the units.

### Example 3: different units

Find the volume of this triangular prism.

Write down the formula.

Calculate the area of the triangular cross-section and substitute the values.

Solve the equation.

Write the answer, including the units.

## How to find a missing length given the volume

1. Write down the formula.
2. Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.
3. Solve the equation.
4. Write the answer, including the units.

## Missing length examples

### Example 4: finding a missing length

The volume of this triangular prism is 168 \mathrm{~cm}^3. Find the length, x, of the triangular prism.

Write down the formula.

Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.

Solve the equation.

Write the answer, including the units.

### Example 5: finding a missing height

The volume of this triangular prism is 80 \mathrm{~mm}^3. Find the height of the prism.

Write down the formula.

Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.

Solve the equation.

Write the answer, including the units.

### Example 6: finding a missing base, different units

The volume of this triangular prism is 440 \mathrm{~mm}^2. Find the length labeled y.

Write down the formula.

Calculate the area of the triangular cross-section and substitute everything into the volume of a triangular prism formula.

Solve the equation.

Write the answer, including the units.

### Teaching tips for volume of a triangular prism

• Find different representations to use during instruction to appeal to all the different learning styles in your classroom. This includes diagrams, equations and verbal cues.

• Find technology tools to incorporate in teaching, including interactive whiteboards and interactive manipulatives.

• Make sure that triangular prism volume worksheets have a variety of problem types to challenge students and reinforce understanding.

### Easy mistakes to make

• Missing units or using incorrect units
Volume is measured in units cubed ( \mathrm{mm}^3, \mathrm{~cm}^3, \mathrm{~m}^3 etc).

• Calculating with different units
You need to make sure all measurements are in the same units before calculating volume. Example, you can’t have some in cm and some in m.

• Using the wrong formula
Be careful to apply the correct prism related formula to the correct question type.

### Practice volume of a triangular prism questions

1. Find the volume of the triangular prism.

110 \mathrm{~cm}^{3}

55 \mathrm{~cm}^{3}

240 \mathrm{~cm}^{3}

120 \mathrm{~cm}^{3}
\begin{aligned} \text{Area of triangle }&=\cfrac{1}{2} \times 3 \times 8 \\\\ &=12\mathrm{~cm}^{2}\end{aligned}

\begin{aligned}\text{Volume of triangular prism }&=12 \times 10 \\\\ &=120\mathrm{~cm}^{3}\end{aligned}

2. Find the volume of the triangular prism.

252 \mathrm{~cm}^{3}

42 \mathrm{~cm}^{3}

1764 \mathrm{~cm}^{3}

126 \mathrm{~cm}^{3}
\begin{aligned}\text{Area of triangle }&=\cfrac{1}{2} \times 6 \times 6 \\\\ &=18\mathrm{~cm}^{2}\end{aligned}

\begin{aligned}\text{Volume of triangular prism }&=18 \times 7 \\\\ &=126\mathrm{~cm}^{3}\end{aligned}

3. Find the volume of the triangular prism.

924 \mathrm{~mm}^{3}

9.24 \mathrm{~cm}^{3}

92.4 \mathrm{~mm}^{3}

9240 \mathrm{~mm}^{3}

Notice that one of the measurements is in mm and the others are in cm. Change 1.4 \mathrm{~cm} to 14 \mathrm{~mm} and 2.2 \mathrm{~cm} to 22 \mathrm{~mm}.

\begin{aligned}\text{Area of triangle}&=\cfrac{1}{2} \times 14 \times 6 \\\\ &=42\mathrm{~mm}^{2}\end{aligned}

\begin{aligned}\text{Volume of triangular prism }&=42 \times 22 \\\\ &=924\mathrm{~mm}^{3}\end{aligned}

4. The volume of this right triangular prism is 84\mathrm{~cm}^{3}.  Find the length, x, of the triangular prism.

3.5\mathrm{~cm}

2,016\mathrm{~cm}

7\mathrm{~cm}

1,008\mathrm{~cm}
\begin{aligned}\text{Area of triangle }&=\cfrac{1}{2} \times 3 \times 8 \\\\ &=12\mathrm{~cm}^{2}\end{aligned}

\begin{aligned}\text{Volume of triangular prism }&=12 \times x \\\\ 84&= 12x \\\\ 7&=x\end{aligned}

The length is 7 \mathrm{~cm}.

5. The volume of this triangular prism is 405\mathrm{~m}^{3}. Find the height of the triangular prism.

45 \mathrm{~m}

9 \mathrm{~m}

2.5 \mathrm{~m}

4.5 \mathrm{~m}
\begin{aligned}\text{Area of triangle }&=\cfrac{1}{2} \times 6 \times h \\\\ &=3h\end{aligned}

\begin{aligned}\text{Volume of triangular prism }&=3h \times 15\\\\405 &= 45h \\\\ 9&=h\end{aligned}

The height is 9 \mathrm{~m}.

6. The volume of this triangular prism is 45\mathrm{~cm}^{3}

Find the length of y.

5 \mathrm{~cm}

0.25 \mathrm{~cm}

0.5 \mathrm{~cm}

4,050 \mathrm{~cm}
\begin{aligned}\text{Area of triangle }&=\cfrac{1}{2} \times 4 \times y \\\\ &=2y\end{aligned}

Notice that the height of the triangular prism is in mm , however the volume is in cm^3. Therefore, you need to change 45 \mathrm{~mm} to 4.5 \mathrm{~cm}.

\begin{aligned}\text{Volume of triangular prism }&=2y \times 4.5\\\\ 45&= 9h \\\\ 5&=h\end{aligned}

The length of y is 5 \mathrm{~cm}.

## Volume of a triangular prism FAQs

What is the difference between a triangular prism and a rectangular prism?

A triangular prism is a 3D shape with triangular bases and rectangular faces or sides with 9 vertices. A rectangular prism is a 3D shape with 6 rectangular sides or faces that meet at right angles with 8 vertices.

Can the triangular base of the prism have different sizes?

In a triangular prism, the bases must be identical triangles, which ensures that the prism has a uniform cross-section throughout its length.

What is Heron’s formula?

Heron’s formula is a formula used to find the area of a triangle in terms of the length of its sides. It can be applied to any type of triangle, including equilateral triangles.

Heron’s formula is A=\sqrt{s(s-a)(s-b)(s-c)}

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