Volume (VCE SSCE General Mathematics): Revision Notes

Volume

What is volume?

Volume measures how much three-dimensional space an object occupies. When you calculate volume, you're finding out how much room something takes up. Volume is always expressed in cubic units, such as mm³ (cubic millimetres), cm³ (cubic centimetres), or m³ (cubic metres).

Understanding cross-sections

Many three-dimensional shapes, particularly prisms and cylinders, have a uniform cross-section along their entire length. A cross-section is the shape you would see if you sliced straight through the object.

For any prism or cylinder, you can find the volume using this fundamental principle:

$\text{Volume} = \text{area of cross-section} \times \text{height (or length)}$

This means you first work out the area of the shape's cross-section, then multiply it by how long or tall the shape is.

Volume formulas for common shapes

Different three-dimensional shapes require different formulas to calculate their volume. Let's look at the most common ones.

Rectangular prism (cuboid)

A rectangular prism, also called a cuboid, is a box-shaped object with rectangular faces. Think of a shoebox or a brick.

$V = lwh$

where:

• $l$ = length
• $w$ = width
• $h$ = height

Square prism (cube)

A cube is a special rectangular prism where all edges are the same length. Think of a dice or a Rubik's cube.

$V = l^3$

where:

• $l$ = length of one edge

This formula is simpler because you're just multiplying the same measurement three times.

Triangular prism

A triangular prism has a triangular cross-section. Think of a Toblerone chocolate bar or a camping tent.

$V = \frac{1}{2}bhl$

where:

• $b$ = base of the triangle
• $h$ = height of the triangle
• $l$ = length of the prism

Cylinder

A cylinder is like a circular prism. Think of a tin can or a pipe.

$V = \pi r^2 h$

where:

• $r$ = radius of the circular base
• $h$ = height of the cylinder

Worked example: Finding the volume of a cuboid

Let's work through calculating the volume of a cuboid step by step.

Worked example: Finding the volume of a cylinder

Cylinders require the use of π in their volume formula.

Finding the volume of composite shapes

Sometimes you'll encounter shapes made up of more than one basic form. For these composite shapes, you need to break the problem down into steps.

Volume of a cone

A cone is a three-dimensional shape that tapers from a circular base to a point (called the apex or vertex).

This gives us the formula:

$V = \frac{1}{3}\pi r^2 h$

where:

• $r$ = radius of the circular base
• $h$ = perpendicular height from base to apex

A right circular cone is one where a line from the centre of the base to the apex is perpendicular to the base. This is the most common type of cone you'll work with.

Volume of a sphere

A sphere is a perfectly round three-dimensional object, like a ball. Every point on its surface is the same distance from its centre.

The volume of a sphere with radius $r$ is:

$V = \frac{4}{3}\pi r^3$

where:

• $r$ = radius of the sphere

Capacity

While volume measures the space an object occupies, capacity measures the amount of substance (usually liquid) that an object can hold.

Volume to capacity conversions

To convert between volume and capacity, use these key relationships:

$1 \text{ m}^3 = 1000 \text{ litres (L)}$

$1 \text{ cm}^3 = 1 \text{ millilitre (mL)}$

$1000 \text{ cm}^3 = 1 \text{ litre (L)}$

Summary of volume formulas

Here's a quick reference table of all the volume formulas you need to know:

Shape	Volume Formula	Where
Rectangular prism	$V = lwh$	$l$ = length, $w$ = width, $h$ = height
Triangular prism	$V = \frac{1}{2}bhl$	$b$ = base, $h$ = height, $l$ = length
Cylinder	$V = \pi r^2 h$	$r$ = radius, $h$ = height
Cone	$V = \frac{1}{3}\pi r^2 h$	$r$ = radius of base, $h$ = height
Sphere	$V = \frac{4}{3}\pi r^3$	$r$ = radius