Volume of regular solids

Before you can say how tightly packed a substance is — its density — you first need a number for how much space it occupies. That number is volume, and for a solid with straight edges and flat faces, like a matchbox, a brick, or a laboratory metal block, finding it takes nothing more than a ruler and multiplication. This section, part of CBSE Class 8 Chemistry's chapter on solutes, solvents and solutions, builds the volume skill that the density formula leans on. You will see the SI and practical units of volume, the length times width times height rule for a cuboid, how a measuring cylinder reads a liquid's volume, and where the ruler-and-multiply method stops working — handing over to the water-displacement method for oddly shaped objects like a stone or a key.

The core insight

The volume of a regular solid such as a cuboid is found by multiplying its three measured dimensions, length times width times height, all expressed in the same unit, giving an answer in a cubic unit such as cm³.

The core idea

Volume is the amount of space an object occupies. Its SI unit is the cubic metre ( $\text{m}^3$ ), but in a school laboratory smaller, more practical units are used instead:

Cubic centimetre ( $\text{cm}^3$ , also written cc) and cubic decimetre ( $\text{dm}^3$ ).
Litres (L) for liquids, where $1\ \text{L} = 1\ \text{dm}^3$ .
Millilitres (mL), where $1\ \text{mL} = 1\ \text{cm}^3$ — a conversion used constantly once solids and liquids are compared.

A regular solid is one whose shape is a simple, named geometric form — a cuboid (box shape) or a cube — with flat faces and straight edges that a ruler can measure directly. For a cuboid of length $l$ , width $w$ , and height $h$ :

\text{Volume} = l \times w \times h

where: $l$ , $w$ , $h$ are the three edge lengths measured in the same unit (for example, all in centimetres), and the resulting Volume is expressed in the cube of that unit (cm³, m³, and so on). A cube is simply the special case where $l = w = h$ . Two things make this formula usable in practice. First, unit consistency: multiplying a length in centimetres by a width in millimetres without converting gives a meaningless number — every dimension must be brought to one unit before the multiplication happens. Second, scope: the formula only applies to solids regular enough to measure with a ruler. A stone, a key, or a crumpled piece of foil has no straight edges to measure, so a different technique — lowering it into a measuring cylinder and reading the rise in water level, the water-displacement method — is used instead (covered as its own topic). Once a solid's volume is known, it becomes the second ingredient (alongside mass) needed for $\text{Density} = \text{Mass}/\text{Volume}$ , the very next idea in this chapter. Liquids are measured differently, with a measuring cylinder — a narrow, graduated glass or plastic vessel sold in standard sizes such as 5, 10, 25, 50, 100, and 250 mL. Reading one accurately means first checking the size of each small division: if a 100 mL cylinder has 10 divisions marked between its 10 mL and 20 mL lines, each division is worth 1 mL, so a reading can be trusted to the nearest 1 mL. Misreading the division size is a common source of error, so this check always comes first.

Figure

A cuboid with its length width and height labelled beside a formula box showing volume equals length times width times height

Derivation AVolume of a cuboid from three ruler measurements

The goal of this derivation is to show, from the basic idea of what a cubic unit means, exactly why multiplying three perpendicular edge lengths together gives the space a cuboid occupies — not just state the formula but justify why length times width times height is the right operation rather than, say, addition or an average of the three edges.

Step 1. Picture the cuboid's base, a rectangle of length $l$ and width $w$ , covered edge-to-edge with unit cubes, each cube exactly $1$ unit on a side. A rectangle of area $l \times w$ (in square units) holds $l \times w$ of these unit cubes laid flat, because area itself is defined as rows of $w$ unit squares repeated $l$ times.

Step 2. Now stack that same layer of $l \times w$ unit cubes upward, one layer per unit of height, until the stack reaches height $h$ . Since each layer contributes $l \times w$ unit cubes and there are $h$ layers, the total count of unit cubes filling the cuboid is $l \times w \times h$ .

Step 3. Each unit cube, by definition, occupies exactly one cubic unit of space (one cm³ if the ruler measured in centimetres). Since the whole solid is filled by $l \times w \times h$ such cubes with no gaps and no overlap, the total space it occupies — its volume — must equal that same count: $\text{Volume} = l \times w \times h$ .

Step 4. This also explains the unit rule seen in the worked examples below: multiplying three lengths in centimetres gives an answer whose unit is $\text{cm} \times \text{cm} \times \text{cm} = \text{cm}^3$ , a cubic unit, because each factor contributed one linear dimension to the three-dimensional stack of unit cubes.

Worked example

A rectangular wooden plank has a length of 25 cm, a width of 18 cm, and a thickness (height) of 2 cm. Find its volume.

Step 1. Write down the given values with their unit already matching: $l = 25\ \text{cm}$ , $w = 18\ \text{cm}$ , $h = 2\ \text{cm}$ . All three are in centimetres, so no conversion is needed before multiplying.

Step 2. Write the formula and substitute: $\text{Volume} = l \times w \times h = 25\ \text{cm} \times 18\ \text{cm} \times 2\ \text{cm}$ .

Step 3. Multiply in two stages to avoid arithmetic slips: $25 \times 18 = 450$ , then $450 \times 2 = 900$ .

Step 4. Attach the cubic unit to the number: $\text{Volume} = 900\ \text{cm}^3$ . The wooden plank occupies a volume of 900 cm³.

Competitive stretch

A metal block used in the same laboratory is a cube with each side measuring 4 cm. Find its volume in cm³, and express the same answer in millilitres. Here $l = w = h = 4\ \text{cm}$ (a cube is simply a cuboid with equal sides), so $\text{Volume} = 4 \times 4 \times 4 = 16 \times 4 = 64\ \text{cm}^3$ . Since $1\ \text{mL} = 1\ \text{cm}^3$ , the same volume written in millilitres is numerically identical: the cube's volume is 64 cm³, which equals 64 mL — no extra calculation is needed once the unit relationship is remembered.

Common traps

Multiplying dimensions that are given in different units — for instance, one edge in cm and another in mm — without converting all three to the same unit first, which silently corrupts the final answer.
Forgetting to cube the unit: the answer to $l \times w \times h$ must carry a cubic unit (cm³, m³), never a plain linear unit like cm.
Applying $l \times w \times h$ to an object that is not truly a regular cuboid or cube — irregular objects like a stone or a key need the water-displacement method, not a ruler.
Mixing up volume with mass: volume is the space occupied, mass is the amount of matter present; they are linked through density but are never interchangeable.

Ask the tutor: Want to try volume problems with mixed units, or see how today's volume answer plugs into a density calculation next?

Why does the formula use multiplication and not just adding the three sides

Because volume measures three-dimensional space, not a one-dimensional total. Imagine filling the cuboid with tiny unit cubes: one layer covering the base holds $l \times w$ cubes, and stacking $h$ such layers gives $l \times w \times h$ cubes in all. Adding the three edges would only ever give a length, never the count of cubes needed to fill the solid, so it cannot represent volume.

Can I use this same formula for a cylinder or a ball

No. The formula $\text{Volume} = l \times w \times h$ is specific to a cuboid or cube, shapes with three straight, mutually perpendicular edges. A cylinder, sphere, or cone has curved surfaces and needs its own formula (built from radius and height, met in later classes), and an object with no regular shape at all — like a stone — needs the water-displacement method instead, since it has no straight edges to put a ruler against.

If a cube has a volume of 64 cm³, why is that the same as 64 mL

Because the millilitre and the cubic centimetre are defined to be exactly equal: $1\ \text{mL} = 1\ \text{cm}^3$ . This is not a coincidence but a deliberate choice in how the units were set up, so a volume measured in cm³ with a ruler can be compared directly to a volume measured in mL with a measuring cylinder, without any conversion factor needed.

Volume of regular solids

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads