We have already looked at finding the amount of space enclosed by a 2D shape - this was called AREA. (Remember doing areas of rectangles, triangles and parallelograms?)

Well what if we have a 3D shape? How can we find the space inside of that? The space enclosed by a 3D shape is called VOLUME.

Exploring Volume

Similar to how Area is measured in square units, Volume is measured in cubic units.

Here is $1$1 cubic unit.

If we join two together we get $2$2 cubic units, add another for $3$3 cubic units and another for $4$4. Got it?

It seems easy to measure for long thin sticks.... what if the cubic units are not arranged in a line?

Here is a $2\times4$2×4 arrangement (that is $1$1 block high). It is made up of $8$8 cubic units.

If I add another row ($2$2 blocks high), it will be made up of $16$16 cubic units. ($8$8 in the first row and $8$8 in the second)

What do you think will happen if I add another row? How many cubic units now?

We have $2\times4$2×4 cubic units on the bottom row, and we have 3 rows, so we have $2\times4\times3$2×4×3 cubic units = $24$24 cubic units.

Have you seen a pattern with how to work out how many cubic units there are in a rectangular solid?

Rectangular Prisms

Rectangular solids like these are called rectangular prisms. To work out the volume of a rectangular prism we multiply the $\text{Length }\times\text{Width }$Length ×Width (of the base) and then multiply that by the number of rows, which is the $\text{Height }$Height .

Volume of Rectangular Prisms

$\text{Volume of Rectangular Prism }=\text{Length }\times\text{Width }\times\text{Height }$Volume of Rectangular Prism =Length ×Width ×Height

$V=L\times W\times H$V=L×W×H

Units

We use special units to describe volume, based on the notion of cubic units described above. Because the units for length include millimetres, centimetres, metres and kilometres we end up with the following units for area.

Units of Volume

cubic millimetres = mm³

(picture a cube with side lengths of $1$1 mm each - pretty small this one!)

cubic centimetres = cm³

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

cubic metres = m³

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

Worked Examples

QUESTION 1

Find the volume of the cube shown.

A three-dimensional cube with edges depicted in a green outline. The front bottom edge of the cube is labeled with the measurement of $12$12 cm.

QUESTION 2

Find the volume of the rectangular prism shown.

A three-dimensional rectangular prism with length dimensions labeled. The height of the prism is labeled as " $4$ cm," the width as " $6$ cm," and the length as " $14$ cm." The prism is outlined in green, and the lines representing its edges are slightly skewed to give a sense of depth.

QUESTION 3

This solid consists of a rectangular prism with a smaller rectangular prism cut out of it. Find the volume of the solid.

Outcomes

5.M2.08

Determine, through investigation using stacked congruent rectangular layers of concrete materials, the relationship between the height, the area of the base, and the volume of a rectangular prism, and generalize to develop the formula (i.e., Volume = area of base x height)

Volume of rectangular prisms