**Learning Goal**

By the end of this lesson I will be able to find the volume of a prism when given its dimensions.

**Volume and Prisms**

The volume of a solid figure is the measure of how much three-dimensional space it takes up or holds.

For example if you imagine a fish tank, the amount of water it contains is its volume. Its length, width, and height determine how much water the tank will hold.

**Example A.** Determine the volume of the following prism assuming that each cube shown is a unit cube. A unit cube has length, width and height measurements that are all one unit.

This prism has three unit cubes lined up for the length, it has two unit cubes along the side for the width and it has four unit cubes lined up for the height. If we count all of these unit cubes, then we can see that we have 24 unit cubes.

**Rectangular Prisms**

The formula for the volume of rectangular prism is a version of what happened in example A. Using the length and the width you are able to find the number of cubes in the first layer of the figure. The height represents the number of layers. Multiplying the area of the base by the height will result in the volume.

Therefore the formula for the volume of a * rectangular prism* is \begin{align*}V = lwh\end{align*}.

**Example B.** Find the volume of the prism below.

We simply put the values for the length, width, and height in for the appropriate variables in the formula. Then we solve for \begin{align*}V\end{align*}, volume.

\begin{align*}Volume = lwh\end{align*}

\begin{align*}Volume=(16)(9)(4) \end{align*}

\begin{align*}Volume =576cm^3\end{align*}

The volume of this rectangular prism is \begin{align*}576 \ cm^3\end{align*}. Notice that the exponent three was used to represent the cubic units of the figure. All volume is measured in cubic units, so you will need to use this exponent when working on figuring out the volume of a solid.

**Triangular Prisms**

The formula for the volume of triangular prism again involves the multiplication of the base area by the number of layers (height). In this case however the base of the figure is a triangle so instead of using \begin{align*}A= (base)(height)\end{align*} you will need to use the formula for the area of a triangle which is \begin{align*}A=\frac{(base)(height)}{2} \end{align*} .

Therefore the formula for the volume of a * triangular prism* is \begin{align*}V=\frac{blh}{2}\end{align*} . The one confusing part of this formula is that there is a height for the entire prism and a height of the triangle. The height of the triangle is given the letter \begin{align*}l\end{align*}, and the height of the entire prism is given the letter \begin{align*}h\end{align*} to make sure all of the values are put into the formula in the right place.

**Example C.** Find the volume of the prism below.

This is a triangular prism with a height of 10cm, a triangle base measurement of 16cm and a triangle height of 6cm. Put these values into the formula to calculate the volume.

\begin{align*}Volume = \frac{blh}{2}\end{align*}

\begin{align*}Volume = \frac{(16)(6)(10)}{2}\end{align*}

\begin{align*}Volume = 480cm^3\end{align*}

**Example D.** Find the volume of the triangular prism.

This triangular prism has a height measurement of 7cm and a base measurement of 8cm. The 12cm however is not the height of the triangle, \begin{align*}l\end{align*}. The value of \begin{align*}l\end{align*} must be found using a right angle triangle and Pythagorean Theorem. The figure has been redrawn below.

To find the value of \begin{align*}l\end{align*}, use Pythagorean Theorem.

\begin{align*}a^2+b^2 = c^2\end{align*}

\begin{align*}4^2 + l^2 = 12^2\end{align*}

\begin{align*}16 + l^2 = 144\end{align*}

\begin{align*}l^2 = 144 - 16\end{align*}

\begin{align*}l^2 = 128\end{align*}

\begin{align*}l=\sqrt{128}\end{align*}

\begin{align*}l=11.31cm\end{align*}

Once the value of \begin{align*}l\end{align*} is found, the volume can be found using the formula.

#### \begin{align*}Volume = \frac{blh}{2}\end{align*}

#### \begin{align*}Volume = \frac{(8)(11.31)(7)}{2}\end{align*}

\begin{align*}Volume = 316.68cm^3\end{align*}

For a review of the concepts covered please watch the following videos.

**Practice**

Directions: Find the volume of each of the following.

a)

(Solution: 1679.29cm^{3})

b) A rectangular prism with a base of 20cm by 15cm and a height of 2cm.

(Solution: 600cm^{3})

c)

(Solution: 198cm^{3})

d) A triangular prism with a base area of 144m^{2} and a height of 10m.

(Solution: 1440m^{2})