Volume of solids

Area says how much room a figure takes up on a plane. Volume says how much fits inside a solid — and we count it in cubes rather than in squares.

Volume is counted cubes

The area of a rectangle is the number of unit squares that cover it. The volume of a cuboid is the number of unit cubes that fill it.

Lay a single layer of $1\,\text{cm}$ cubes on the bottom: it takes $a \cdot b$ of them — exactly the base area. There are $c$ such layers. Hence:

A cube is a cuboid with equal edges, so its formula is the same formula:

The shared rule: base area times height

The layer argument works for every solid with a constant cross-section: the base area says how much one layer holds, the height says how many layers there are.

For a cuboid $A_b = a \cdot b$. For a cylinder the base is a circle, so $A_b = \pi r^2$:

Pyramids and cones: one third

A solid that narrows to an apex has no constant cross-section. It turns out to hold exactly three times less than the straight solid on the same base and height:

Three identical pyramids sharing a base and a height fit together into one prism — the same experiment you can run in a kitchen by pouring water from a conical funnel into a cylindrical glass: it has to be filled three times.

A cone is in exactly the same position — its base is a circle, so

The sphere

A sphere has neither a base nor a height — only a radius. Its formula is the only one here in which the radius appears to the third power:

Units

Volume comes from multiplying three lengths, so its unit is a length unit multiplied three times: $\text{cm} \cdot \text{cm} \cdot \text{cm} = \text{cm}^3$. That is why the answer is written in cubic centimetres or cubic metres — never in $\text{cm}^2$, which is a unit of area.

A litre is another name for the same quantity: $1\,\text{l} = 1\,\text{dm}^3$. If you need actual values converted — litres to millilitres, cubic metres to litres, gallons to litres — our volume converter will do it.

Exercises

If the question does not name a unit, type it together with the answer, e.g. 60 cm³. If the unit is given in brackets (because $\pi$ shows up in the answer), type the number alone, rounded to two decimal places. An answer worked out with $\pi \approx 3.14$ is accepted too.

Common mistakes

• A volume in square units — $60\,\text{cm}^3$, not $60\,\text{cm}^2$; the square belongs to area.
• The lost one third — a pyramid and a cone carry $\frac{1}{3}$; without it the answer is three times too big.
• A lateral edge instead of the height — the formula takes the segment perpendicular to the base, not the slanted one.
• The diameter instead of the radius — a cylinder, a cone and a sphere all take $r$; the diameter makes the answer four or eight times too big.
• $(\pi r)^2$ instead of $\pi r^2$ — only the radius is squared.
• $r^3$ read as $3r$ — the third power is $r \cdot r \cdot r$, not a multiplication by three.
• Dimensions in mixed units — bring them all to one unit before multiplying.