7.05 Volume of prisms

Remember that a quantity of volume is represented in terms of the volume of a unit cube, which is a cube with side length 1 unit. By definition, a single unit cube has a volume of 1 cubic unit, written as 1\text{ unit}^3.

We use special units to describe volume, based on the notion of cubic units described above. Because the units for length include millimeters, centimeters, meters and kilometers we end up with \text { mm}^3, \text { cm}^3, and \text { m}^3 for example.

The volume of a three dimensional shape is the amount of space that is contained within that shape.

In the same way that the area of a two dimensional shape is related to the product of two perpendicular lengths, the length and width, the volume of a three dimensional shape is related to the product of three perpendicular lengths, the length, width, and height. Notice that each of the three lengths is perpendicular to the other two.

The volume of a rectangular prism is given by

\begin{aligned} \text{Volume }&=\text{length }\times \text{width }\times \text {height,\quad}\text{or}\\ V&=l\times w\times h \end{aligned}

Examples

Example 1

Find the volume of the rectangular prism shown.

Worked Solution

Create a strategy

Use the volume of a rectangular prism formula.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle l\times w\times h	Use the volume formula
	\displaystyle =	\displaystyle 15\times4\times2	Substitute l=15, w=4, and h=2
	\displaystyle =	\displaystyle 120\text{ cm}^3	Evaluate

Watch question walkthrough

Recall that prisms have rectangular sides, and the shape on the top and the base can be a variety of other polygons.

We can find the area of other types of prisms that are not rectangular, by finding the area of one base, and then multiplying that by the height of the prism. Let's look at a worked example to see how.

Examples

Example 2

Determine the volume of the prism in cubic centimeters.

Worked Solution

Create a strategy

Find the area of the base, then multiply by the height of the prism.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times\text h	Use the formula for the area of a triangle
	\displaystyle =	\displaystyle \dfrac12\times4\times3	Substitute b=4 and h=3
	\displaystyle =	\displaystyle 6	Evaluate
\displaystyle V	\displaystyle =	\displaystyle 6\times 6	Multiply the area of the base by the height of the prism
	\displaystyle =	\displaystyle 36\text{ cm}^3	Evaluate

Example 3

Find the volume of the following prism:

Worked Solution

Create a strategy

Find the volume of the prism by decomposing the trapezoid base into a triangle and a rectangle and adding the areas, then multiply by the height of the prism.

Apply the idea

Let b=3 and h=7 to find the area of the triangle and l=10 and w=7 to find the area of the rectangle.

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times \left(b \times h\right) + l\times w	Use the formula for the area of a triangle and rectangle
	\displaystyle =	\displaystyle \dfrac12\times(3\times 7)+ 10 \times 7	Substitute the values of b, h, l and w
	\displaystyle =	\displaystyle 10.5 + 70	Evaluate the multiplication
	\displaystyle =	\displaystyle 80.5	Evaluate the addition
\displaystyle V	\displaystyle =	\displaystyle 80.5\times 5	Multiply the area of the base by the height of the prism
	\displaystyle =	\displaystyle 402.5	Evaluate

The volume of the prism is 402.5\text{ cm}^3