7. Geometry

Lesson

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

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

The image below shows a rectangular prism with length $5$5 units, width $3$3 units, and height $2$2 units. Notice that the length of each edge corresponds to the number of unit cubes that could be lined up side by side along that edge.

We can find the number of unit cubes that could fit inside the rectangular prism by taking the product of the three side lengths. This gives $5\times3\times2=30$5×3×2=30, so there are $30$30 unit cubes in the prism, which means it has a volume of $30$30 unit^{3}.

Use the sliders to change the length, width, and height of the rectangular prism. Consider the questions below.

- Why do you think all of the unit cubes in the base are shown?
- If we count the number of unit cubes in the base, how can we use the height to get the total volume (number of unit cubes)?
- What product could we use the find the volume?

In the same way that the area of a two dimensional shape is related to the product of two perpendicular lengths, the volume of a three dimensional shape is related to the product of three mutually perpendicular lengths (each of the three lengths is perpendicular to the other two).

Volume of a rectangular prism

The volume of a rectangular prism is given by

$\text{Volume }=\text{length }\times\text{width }\times\text{height }$Volume =length ×width ×height , or

$V=l\times w\times h$`V`=`l`×`w`×`h`

Use the three sliders for length, width, and height to see how changing these affect the rectangular prism. Click the boxes to see the formula and volume revealed.

A cube can be thought of as a special type of rectangular prism, one that has all sides equal in length. The formula for the volume of a cube is similar to the formula for the area of a square.

Volume of a cube

The volume of a cube is given by

$\text{Volume }=\text{side }\times\text{side }\times\text{side }$Volume =side ×side ×side , or

$V=s\times s\times s=s^3$`V`=`s`×`s`×`s`=`s`3

Find the volume of the following rectangular prism.

**Think**: The side lengths have units of cm, so the volume will be in cm^{3}.

**Do**: The base of the prism has a width of $2$2 cm and a length of $7$7 cm, and the height of the prism is $9$9 cm. We will use these sides in the formula for the volume of a rectangular prism.

$\text{Volume }$Volume | $=$= | $\text{length }\times\text{width }\times\text{height }$length ×width ×height | (Formula for the volume of a rectangular prism) |

$=$= | $7\times2\times9$7×2×9 | (Substitute the values for the length, width, and height) | |

$=$= | $126$126 | (Perform the multiplication to find the volume) |

So this rectangular prism has a volume of $126$126 cm^{3}.

The local swimming pool is $25$25 m long. It has eight lanes, each $2$2 m wide, and its depth is $1.5$1.5 m. What is the volume of water in the pool?

**Think**: The water in the pool is in the shape of a rectangular prism, so to find its volume we need to find the side lengths of this prism. The length and depth of the pool are two side lengths we can use. The final side length is found by multiplying the number of lanes by the width of each lane.

**Do**: First we calculate the width of the pool using the width of each swim lane: $8\times2$8×2 m $=16$=16 m. Next we use the formula for the volume of a rectangular prism.

$\text{Volume }$Volume | $=$= | $\text{length }\times\text{width }\times\text{height }$length ×width ×height | (Formula for the volume of a rectangular prism) |

$=$= | $25\times16\times1.5$25×16×1.5 | (Substitute the values for the length, width, and height) | |

$=$= | $600$600 | (Perform the multiplication to find the volume) |

So the water in the pool has a volume of $600$600 m^{3}.

**Reflect**: Even though the volume formula uses the terms "length", "width", and "height", when referring to everyday objects it may be more appropriate or more common to use alternative words like "width", "depth", or "thickness". In this example, we could just as well have used the formula $\text{Volume }=\text{length }\times\text{width }\times\text{depth }$Volume =length ×width ×depth .

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 the following units for area.

Units of Volume

**cubic millimeters = mm ^{3}**

(picture a cube with side lengths of $1$1 mm each - that's pretty small!)

**cubic centimeters = cm ^{3}**

(picture a cube with side lengths of $1$1 cm each - about the size of a dice)

**c****ubic meters = m ^{3}**

(picture a cube with side lengths of $1$1 m each - what could be this big?)

Before we start a question, it is important to check that all of the sides are in the **same unit**. If they aren't, then we should convert them to the same unit.

Find the volume of the rectangular prism shown.

Find the volume of the cube shown.

A box is $1$1 m long, $20$20 cm high and $30$30 cm wide. What is the volume of the box in cubic centimeters?

Describe the two-dimensional shapes that result from slicing three-dimensional solids parallel or perpendicular to the base.

Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids. Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids. Find the volume of right triangular prisms, and solve volume problems involving three- dimensional objects composed of right rectangular prisms.