Volume

Study Sheet

Volume

Filling a box with unit cubes

Counting Cubes

Concept
The Big Idea

Volume counts the unit cubes that fill a solid. One layer holds length×width\text{length} \times \text{width} cubes, and there are height\text{height} layers.

Volume counts how many unit cubes fill a solid. For a box, a layer of cubes on the bottom has length ×\times width cubes, and there are height-many layers, so V=ℓ×w×hV = \ell\times w\times h. Units are cubed — cubic centimeters, cubic feet — because three lengths were multiplied.

Tip
The formula

For a rectangular box, V=l×w×hV = l \times w \times h, measured in cubic units.

Example
A 4 by 3 by 2 box
432
Each layer is 4 × 3 = 12 cubes; there are 2 layers.

One layer is 4×3=124 \times 3 = 12 cubes, and there are 22 layers: 2424 cubic units.

Example
Stacking layers
334
Four layers of 9.

A box has a 33 by 33 base and 44 layers. One layer is 3×3=93 \times 3 = 9 cubes. Four layers hold 9×4=369 \times 4 = 36 cubes, so the volume is 3636 cubic units.

Side note
Where it goes wrong

Mixing units, like a length in centimeters and a height in meters, or reporting a volume in square units. Convert everything to one unit first, then cube it.

Try it
Try it: count the cubes

A box is 55 cubes long, 22 wide and 44 tall. How many unit cubes fill it? If each cube is 11 cm on a side, what is the volume?

Show the answer

One layer: 5×2=105 \times 2 = 10 cubes. Four layers: 10×4=4010 \times 4 = \mathbf{40} cubes. Each cube is 11 cm3^3, so the volume is 40\mathbf{40} cm3^3.

In Plain Terms & More Examples

Concept
In plain terms

Volume counts the unit cubes that fill a box: multiply length by width by height. The answer is in cubic units.

Example
A 5 by 2 by 3 box
523
Three layers of 5 × 2 = 10 cubes.

V=5×2×3=30V = 5 \times 2 \times 3 = 30 cubic units.

Example
A 6 by 4 by 2 box

V=6×4×2=48V = 6 \times 4 \times 2 = 48 cubic units.

Side note
Where it goes wrong

Adding the edges instead of multiplying. 5+2+3=105 + 2 + 3 = 10 is not the volume of a 55 by 22 by 33 box: the bottom layer alone holds 5×2=105 \times 2 = 10 cubes, and there are 33 layers. Volume =5×2×3=30= 5 \times 2 \times 3 = 30.

Try it
Try it: any edge can be the height

Find the volume of a 77 by 22 by 33 box. Does it matter which edge you call the height?

Show the answer

7×2×3=427 \times 2 \times 3 = \mathbf{42} cubic units. It does not matter: 2×3×72 \times 3 \times 7 and 3×7×23 \times 7 \times 2 are also 4242. Any face can be the base; the layers just stack a different way.

Solids Made of Two Boxes

Concept
Split, then add

An L-shaped solid is two boxes glued together. Find each box's volume and add — the same idea as the area model for multiplication.

Volume counts cubes, and counting the cubes in two separate piles and adding gives the same total as counting them all at once, as long as no cube is in both piles. That is why the split must not overlap.

Example
An L-shape
6423 deep
The dashed line splits the L into a 6 × 2 × 3 box and a 2 × 4 × 3 box on top.

A 6×2×36\times2\times3 box next to a 2×4×32\times4\times3 box: 36+24=6036 + 24 = 60 cubic units.

Example
Taking a piece away
545 deep
Subtracting the missing notch is quicker than adding the pieces.

A block 55 long, 55 deep and 44 tall has a 22 by 22 notch cut out of a top corner, all the way from front to back. Whole block: 5×5×4=1005 \times 5 \times 4 = 100. Notch: 2×2×5=202 \times 2 \times 5 = 20. What is left: 100−20=80100 - 20 = 80 cubic units. Splitting into two boxes agrees: 3×5×4+2×5×2=60+20=803 \times 5 \times 4 + 2 \times 5 \times 2 = 60 + 20 = 80.

Side note
Where it goes wrong

Counting the shared part twice. Split the L-shape into a 6×2×36 \times 2 \times 3 bottom and the whole left column, 2×6×32 \times 6 \times 3, and the 2×2×32 \times 2 \times 3 corner where they meet is in both: 36+36=7236 + 36 = 72 instead of 6060. Pieces must not overlap; here 72−12=6072 - 12 = 60.

Try it
Try it: the missing side

A box has volume 180180 cm3^3, length 99 cm and width 44 cm. What is its height?

Show the answer

Base area 9×4=369\times4 = 36; height =180÷36=5= 180 \div 36 = \mathbf{5} cm.

Word Problems

Concept
Filling things up

Volume answers "how much fits inside?". For liquids, 10001000 cm3^3 is exactly 11 litre, because a litre is the volume of a cube 1010 cm on each side: 10×10×10=100010 \times 10 \times 10 = 1000.

That link turns a box's measurements into how much water it holds: find the volume in cubic centimetres, then divide by 10001000 to get litres.

Example
A fish tank
50 cm20 cm30 cm
Length, width and height of the tank.

A tank is 5050 cm long, 2020 cm wide and 3030 cm tall: 50×20×30=30,00050\times20\times30 = 30{,}000 cm3^3, which is 3030 liters (since 10001000 cm3=1^3 = 1 L).

Example
Cubes of side 2
8 cm6 cm4 cm
Each small cube in the picture is 2 cm on a side.

How many 22 cm cubes fit in a box 88 cm by 66 cm by 44 cm? Along each edge: 8÷2=48 \div 2 = 4, 6÷2=36 \div 2 = 3, 4÷2=24 \div 2 = 2. So 4×3×2=244 \times 3 \times 2 = 24 cubes. Dividing volumes gives the same here: the box is 8×6×4=1928 \times 6 \times 4 = 192 cm3^3, each small cube is 2×2×2=82 \times 2 \times 2 = 8 cm3^3, and 192÷8=24192 \div 8 = 24.

Side note
Where it goes wrong

Forgetting to change units. A pool 33 m by 22 m filled 5050 cm deep does not hold 3×2×50=3003 \times 2 \times 50 = 300 cubic metres. Change 5050 cm to 0.50.5 m first: 3×2×0.5=33 \times 2 \times 0.5 = 3 m3^3.

Try it
Try it: sandbox

A sandbox is 22 m by 1.51.5 m and is filled to a depth of 0.20.2 m. How many cubic meters of sand is that?

Show the answer

2×1.5×0.2=0.62\times1.5\times0.2 = \mathbf{0.6} m3^3.

Base Area Times Height

Concept
Layers again

The number of cubes in one layer is the area of the base, BB. Multiply it by the number of layers, the height: V=B×hV = B \times h. For a box this is the same as ℓ×w×h\ell \times w \times h, because B=ℓ×wB = \ell \times w.

Writing it as B×hB \times h helps when you are told the base area instead of its sides. It is also the form that later works for every prism and cylinder, since all of them are stacks of identical layers.

Example
Base area given

A box has a base of area 2020 cm2^2 and is 77 cm tall. V=20×7=140V = 20 \times 7 = 140 cm3^3. You never needed the base's length and width.

Example
Doubling the height
434
Four layers of 12 instead of two.

A 4×3×24 \times 3 \times 2 box holds 2424 cubes. Make it 44 tall instead: 4×3×4=484 \times 3 \times 4 = 48, twice as much, because there are twice as many layers of 1212.

Example
Same volume, different boxes

2424 unit cubes can make a 2×3×42 \times 3 \times 4 box, a 1×4×61 \times 4 \times 6 box or a 2×2×62 \times 2 \times 6 box: each product is 2424. Boxes with equal volumes can look very different.

Side note
Where it goes wrong

Using the perimeter of the base instead of its area. For a 44 by 33 base, 4+3+4+3=144 + 3 + 4 + 3 = 14 is the distance around the bottom, not the number of cubes on it. The layer holds 4×3=124 \times 3 = 12 cubes.

Try it
Try it: tray or box?

Which holds more: a tray 1010 cm by 1010 cm by 22 cm, or a box 55 cm by 55 cm by 88 cm?

Show the answer

Tray: 10×10×2=20010 \times 10 \times 2 = \mathbf{200} cm3^3. Box: 5×5×8=2005 \times 5 \times 8 = \mathbf{200} cm3^3. They hold exactly the same, even though the tray looks bigger from above.

Why It Works

The bottom layer of a 4×3×24\times3\times2 box is a 4×34\times3 rectangle of cubes — 1212 cubes, by the same reasoning as area. Stacking 22 identical layers gives 12×2=2412\times2 = 24 cubes. That is the whole argument: volume of a box == (area of the base) ×\times (height). The same formula works for any prism or cylinder, because they are stacks of identical layers too — which is why a cylinder's volume is πr2h\pi r^2h.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.