Volume & Surface AreaCambridge IGCSE Maths: Revision notes

Section 1

What formulas do I need for volume?

All formulas below are given in the exam's list of formulas, but you must know when to use each:

Solid	Volume formula
Prism	$V = Al$ (cross-sectional area × length)
Cylinder	$V = \pi r^2 h$
Pyramid	$V = \frac{1}{3}Ah$ (base area × height ÷ 3)
Cone	$V = \frac{1}{3}\pi r^2 h$
Sphere	$V = \frac{4}{3}\pi r^3$

A cuboid is a special prism where $A =$ length × width, so $V = lwh$ . A cylinder is also a prism where the cross-section is a circle.

Key termsprismcross-sectional area

Exam tip

For any prism, always find the cross-sectional area first, then multiply by the length — this works for triangular, trapezium-based and L-shaped prisms too.

Section 2

What formulas do I need for surface area?

Curved surface areas (given in the exam):

Cylinder: $2\pi rh$
Cone: $\pi rl$ (where $l$ is the slant height, not the vertical height)
Sphere: $4\pi r^2$

To find the total surface area, add the curved surface area to the area of any flat faces (e.g. a cylinder's total surface area = curved surface area + area of the two circular ends).

Key termscurved surface areaslant heighttotal surface area

Common mistake

Using the vertical height instead of the slant height in the cone surface area formula (πrl) is a very common error — check which one the question gives.

Section 3

How do I approach compound solids and parts of solids? (Extended)

A compound solid is made from two or more basic solids joined together (e.g. a cone stacked on a cylinder). A frustum is what remains when the top of a cone or pyramid is cut off.

Method for compound solid volume:

Identify each basic solid within the compound shape
Calculate the volume of each part separately
Add (or subtract, for a hollow/cut-out shape) the volumes together

Frustum volume = volume of the original cone/pyramid − volume of the smaller cone/pyramid removed from the top.

Key termscompound solidfrustum

Example

A toy is a hemisphere (radius 3 cm) sitting on a cylinder (radius 3 cm, height 5 cm). Total volume = cylinder volume + hemisphere volume = π(3)²(5) + (1/2)(4/3)π(3)³.

Section 4

How do I convert between units of volume and capacity?

Volume units convert using the cube of the linear conversion factor, since volume has three dimensions.

$1 \text{ m} = 100 \text{ cm}$ , so $1 \text{ m}^3 = 100^3 = 1\,000\,000 \text{ cm}^3$
Capacity: $1 \text{ litre} = 1000 \text{ cm}^3$ , and $1000 \text{ litres} = 1 \text{ m}^3$
$1 \text{ ml} = 1 \text{ cm}^3$

Key termscapacity

Common mistake

Forgetting to cube the linear conversion factor for volume (unlike area, which is squared) leads to answers that are wildly out.

Section 5

How do I use net measurements to find volume and surface area?

A net's labelled dimensions directly supply the values needed for both formulas:

For surface area: identify and sum the area of every individual face shown in the net
For volume: identify the base/cross-section from the net and its perpendicular height, then apply the correct volume formula

Always double-check which measurement in a net is the height needed for volume versus the slant length needed for a curved surface area.

Must Know

Prism: V = Al; Cylinder: V = πr²h; Pyramid: V = (1/3)Ah; Cone: V = (1/3)πr²h; Sphere: V = (4/3)πr³
Curved surface areas: cylinder 2πrh, cone πrl (slant height!), sphere 4πr²
Total surface area = curved surface area + any flat face areas
Volume units convert using the cube of the linear factor (1 m³ = 1 000 000 cm³)
1 litre = 1000 cm³, and 1 ml = 1 cm³
Frustum volume = larger cone/pyramid volume − smaller cone/pyramid volume removed

That's the notes covered.

Carry on to the next subtopic.