Mathematics Year 6: Volume of a cuboid and of a cube
Mathematics, Year 6 (Portugal): Fill a box layer by layer and count the unit cubes: length × width × height, and for a cube, edge × edge × edge.
Count the cubes by layers
Fill a cuboid with unit cubes. The bottom layer has length × width cubes, one for each square of the base. Then you stack as many layers as the height says. So V = length × width × height, which is the area of the base times the height. A cube is a cuboid whose three dimensions are equal to the edge a, so V = a × a × a = a³, read “a cubed”.
Multiplying replaces counting
You do not need to fill the box: a layer is a rectangle of cubes, and the same layer repeats. Counting all cubes one by one and multiplying give the same result, and multiplying is much faster. It also explains the units: a cube with a 1 dm edge is 10 cm long, wide and high, so it holds 10 × 10 × 10 = 1000 cubes of 1 cm³. That is why 1 dm³ = 1000 cm³, and the same cube holds 1 l.
A box, a cube and a fish tank
Box 5 cm by 3 cm by 4 cm: the base layer has 5 × 3 = 15 cubes, and there are 4 layers, so V = 15 × 4 = 60 cm³. Cube with edge 6 cm: V = 6 × 6 × 6 = 216 cm³. Fish tank 8 dm by 4 dm by 5 dm: V = 8 × 4 × 5 = 160 dm³, and since 1 dm³ = 1 l, it holds 160 l when full. Keep the same unit for all three dimensions before you multiply.
a³ is not 3 × a
The little 3 in a³ says the edge is multiplied by itself three times, not that it is multiplied by 3. For a cube with edge 6 cm, 6 × 3 = 18 is wrong, and 6 × 6 = 36 is only the area of one face, in cm². The volume is 6 × 6 × 6 = 216 cm³. Watch the units too: a length is in cm, an area in cm², a volume in cm³.
Packing, filling and stacking
Anyone who packs a moving box, fills a tank or a paddling pool, or stacks crates on a pallet is working with the volume of a cuboid. If a crate is 4 dm by 3 dm by 2 dm, it takes up 24 dm³, and you can tell how many fit in a bigger space by comparing volumes. Sugar cubes, dice and building blocks are small cubes where a³ appears by itself.
Test what you learned
A cuboid has a base with sides a and b, and height h, all in the same unit. Which expression gives its volume?
- a) a + b + h
- b) a × b × h, the area of the base times the height
- c) a × b
- d) 2 × (a × b + a × h + b × h)
Correct answer: b) a × b × h, the area of the base times the height — Yes: a × b cubes make one layer, and h layers are stacked, so V = a × b × h.
A cuboid tank is 6 dm long, 5 dm wide and 3 dm high. How many litres of water fill it completely?
- a) 30 l
- b) 900 l
- c) 14 l
- d) 90 l
Correct answer: d) 90 l — Yes: 6 × 5 × 3 = 90 dm³, and since 1 dm³ = 1 l, the tank holds 90 l.
What is the volume of a cube with an edge of 7 cm?
- a) 49 cm³
- b) 21 cm³
- c) 343 cm³
- d) 294 cm³
Correct answer: c) 343 cm³ — Yes: 7 × 7 × 7 = 343. A cube is a cuboid with three equal dimensions.
Keep exploring
Other languages
Loading MyLeoNes™…