Surface Area and Volume
Most of the formulae here are given to you in the exam, which changes what is being tested: not recall, but choosing the right formula, using the right dimension, and handling composite shapes. Almost every recorded error came from one of those three, plus units.
1. The formulae
Prisms and cylinders:
Volume of any prism = cross-sectional area × length
| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | l × w × h | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | ⅓πr²h | πr² + πrl (l = slant height) |
| Sphere | ⁴⁄₃πr³ | 4πr² |
| Pyramid | ⅓ × base area × h | base + triangular faces |
| Hemisphere | ⅔πr³ | 2πr² + πr² (curved + flat) |
Check your formula list. Tutors repeatedly reassured students that the mensuration formulae are provided in the exam — so the marks are for applying them. Confirm what your own paper supplies.
Where the cylinder’s surface area comes from: two circles (2πr²) plus the curved surface, which unrolls into a rectangle of height h and width equal to the circumference (2πr). Understanding that makes it impossible to confuse with the volume.
πr²h is the volume; 2πr² + 2πrh is the surface area. Confusing these two was a documented error — one gives cubic units, the other square units.
2. Radius, diameter and the right height
Formulae use the RADIUS. Questions often give the DIAMETER.
Halve it first. Using the diameter in πr² makes the answer four times too big — and it is the single most common error in the topic.
Square the radius, don’t double it. πr² with r = 5 is π × 25, not π × 10. “Forgot to square the radius” appeared repeatedly.
Cone: slant height vs perpendicular height
The volume uses the perpendicular height h. The curved surface area uses the slant height l.
They are linked by Pythagoras: l² = r² + h²
Questions frequently give one and expect you to find the other. Read carefully which the formula needs.
Hemisphere: its height equals its radius — useful when a hemisphere sits on top of another solid.
3. Composite solids
Shapes made of two or more parts — a cylinder with a cone on top, a cuboid with a hemisphere scooped out.
Volume — simply add or subtract:
Example: a cylinder (r = 3, h = 10) with a hemisphere (r = 3) on top.
- Cylinder: π(3²)(10) = 90π
- Hemisphere: ⅔π(3³) = 18π
- Total = 108π ≈ 339 cm³ (3 s.f.)
Surface area — much more care needed:
Do NOT include faces that are joined together. When a hemisphere sits on a cylinder, the circle where they meet is inside the solid — it is not part of the surface.
For that example:
- Cylinder curved surface: 2π(3)(10) = 60π
- Cylinder base circle: π(3²) = 9π
- Hemisphere curved surface: 2π(3²) = 18π
- The top circle of the cylinder is covered — excluded
- Total = 87π ≈ 273 cm²
Sketch the solid and tick off each surface you can actually see or touch. Including hidden joins, or forgetting the base, were both recorded errors — and they pull the answer in opposite directions, so they don’t cancel out.
“Open” containers have no lid. A cylindrical tank open at the top has surface area πr² + 2πrh — one circle, not two.
4. Working backwards
Given a volume or surface area and asked for a missing dimension:
- Write the formula
- Substitute what you know
- Rearrange and solve
Example: a cylinder of volume 500 cm³ and radius 4 cm. Find the height.
- 500 = π(4²)h = 16πh
- h = 500 ÷ (16π) = 9.95 cm (3 s.f.)
Example: a sphere of volume 288π cm³. Find the radius.
- 288π = ⁴⁄₃πr³ → r³ = 216 → r = 6 cm
Undo the operations in reverse order, and remember a cube root at the end for a sphere or a square root for an area.
A negative volume means an arithmetic error, not a valid answer — check your signs and your subtraction order in composite shapes.
5. Units — where easy marks disappear
Length: cm. Area: cm². Volume: cm³. Always state them.
Converting is not a simple factor of 10:
| Conversion | Factor | Why |
|---|---|---|
| 1 m = 100 cm | ×100 | |
| 1 m² = 10 000 cm² | ×100² | |
| 1 m³ = 1 000 000 cm³ | ×100³ | |
| 1 cm³ = 1 ml | ||
| 1000 cm³ = 1 litre | ||
| 1 m³ = 1000 litres |
Area conversions square the factor; volume conversions cube it. Using ×100 for cm² to m² instead of ×10 000 was a specific documented error.
Convert all measurements to the same unit BEFORE calculating. Mixing metres and centimetres inside one formula was recorded more than once, and the answer is then wrong by a factor of hundreds.
Watch mass units too — grams and kilograms were confused in density questions.
6. Method and accuracy
Keep π in your working and only evaluate at the end, or use the π key. Substituting 3.14 early loses accuracy; some questions want the answer in terms of π anyway, in which case leave it as 108π.
Don’t round intermediate values. In multi-step composite problems, rounding each part before adding is a reliable way to lose the accuracy mark.
Show the substitution. Writing “V = π × 3² × 10” earns method marks even if the arithmetic then slips.
Check your answer is sensible. A volume smaller than one of its parts, or a surface area larger than a solid twice the size, signals an error.
7. Mistakes that cost marks
Using the diameter instead of the radius.
Doubling the radius instead of squaring it.
Confusing the volume and surface area formulae.
Using slant height for volume, or perpendicular height for curved surface area.
Including joined faces in a composite surface area.
Forgetting the base, or including a lid on an open container.
Converting area or volume units with a linear factor.
Mixing units within one calculation.
Omitting units, or using cm² for a volume.
Rounding too early.
Forgetting the cube root when working backwards from a volume.
Frequently asked questions
What is the volume of a prism? Cross-sectional area × length.
What is the volume of a cylinder? πr²h. Its surface area is 2πr² + 2πrh.
What is the volume of a cone? ⅓πr²h, using the perpendicular height.
What’s the difference between slant height and perpendicular height? The slant height l runs up the sloping surface and is used for curved surface area; the perpendicular height h is used for volume. They’re linked by l² = r² + h².
What is the volume of a sphere? ⁴⁄₃πr³, and its surface area is 4πr².
How do I find the surface area of a composite solid? Add only the faces on the outside — exclude any surfaces where the parts join.
How many cm³ are in a m³? 1 000 000 — the conversion factor is cubed.
How do I convert cm³ to litres? Divide by 1000.
How do I find a missing dimension? Substitute into the formula and rearrange, remembering square or cube roots.
Do I need to memorise the formulae? Most are given in the exam — but check your own formula list.
Quick revision checklist
- I know which formulae are provided in my exam
- I always check whether I’m given radius or diameter
- I square the radius rather than doubling it
- I can tell the volume formula from the surface area formula
- I use perpendicular height for volume, slant height for curved surface area
- I can link them with Pythagoras
- I can find volumes of composite solids by adding or subtracting
- I exclude joined faces from composite surface areas
- I check for open tops and missing bases
- I can work backwards to find a missing dimension
- I convert area units by squaring and volume units by cubing
- I convert to a common unit before calculating
- I keep π until the end, or give answers in terms of π
- I give units, correctly squared or cubed
These notes cover surface area and volume in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
Finished this topic?
Saved on this device — no account needed.