What is Volume?

Volume

The amount of three-dimensional space a solid object occupies. Volume is measured in cubic units, such as cm³, m³, or mm³.

Imagine filling a box completely with tiny 1 cm cubes — the total number of cubes needed is the volume of the box in cm³. Volume tells us how much stuff fits inside a shape, or how much space the shape takes up.

Key units to remember:

Cubic centimetres (cm³) — small everyday objects
Cubic metres (m³) — rooms, tanks, large structures
Litres (L) — liquids (1 litre = 1000 cm³); note that litres are commonly used in science but are not a formal SI unit
Millilitres (mL) — 1 mL = 1 cm³

Note

When converting between volume units, remember that lengths are cubed. So 1 m = 100 cm, but 1 m³ = 1,000,000 cm³ (100³). This trips up many students!

Volume in science: Volume isn't just a maths concept — it's central to biology (how large can a cell grow before nutrients can't reach its centre?), chemistry (how much reactant fits in a vessel?), and environmental science (how much water does a reservoir hold?). Throughout these notes, we'll connect the mathematics to real scientific phenomena.

Volume of Prisms and Cylinders (Uniform Cross-Section Solids)

Uniform Cross-Section Solid

A solid where every slice taken parallel to the base produces exactly the same shape and size. Prisms and cylinders are the main examples.

For any solid with a uniform cross-section:

$V = Area of cross-section \times height (or length)$

This single formula covers a huge range of shapes:

Shape	Cross-section	Volume Formula
Rectangular prism (cuboid)	Rectangle	$V = l \times w \times h$
Triangular prism	Triangle	$V = \frac{1}{2} \times b \times h_{△} \times l$
Cylinder	Circle	$V = π r^{2} h$

Note: In the triangular prism formula, $b$ and $h_{△}$ are the base and perpendicular height of the triangular cross-section, while $l$ is the length (depth) of the prism.

Volume of Prisms and Cylinders (Uniform Cross-Section Solids)

Analogy

Think of a stack of identical coins. Each coin is the cross-section, and the height of the stack is the height of the solid. Volume = (area of one coin) × (number of coins in the stack).

Example

Example: Volume of a Rectangular Prism (Cuboid)

A gold ingot measures 10 cm by 5 cm by 2 cm. Find its volume.

Solution:
$V = l \times w \times h = 10 \times 5 \times 2 = 100 cm^{3}$

Example

Example: Volume of a Cylinder

A cylindrical water tank has a base radius of 2 m and a height of 3 m. Find its volume.

Solution:
$V = π r^{2} h = π \times 2^{2} \times 3 = 12 π \approx 37.7 m^{3}$

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