Overview
What you’ll learn
- Convert fluently between metric length, area, volume and capacity units, keeping track of the squared and cubed factors.
- Find perimeters and areas of rectangles, triangles, parallelograms, trapeziums and compound plane shapes.
- Calculate circumference, area, arc length and sector area, quoting exact multiples of π or rounding to 3 significant figures.
- Find volumes and surface areas of prisms, cylinders, pyramids, cones, spheres and compound solids, including parts of solids (Extended).
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 5.1Use metric units of length (mm, cm, m, km), mass (g, kg, tonne), time, area (mm² to km²), volume (mm³ to m³) and capacity (ml, litres); convert between them, including area units such as cm² ↔ m² and volume ↔ capacity such as m³ ↔ litres.
- 5.2Carry out calculations of perimeter and area of rectangles, triangles, parallelograms and trapeziums (only the triangle area formula is given on the formula sheet).
- 5.3Calculate the circumference and area of a circle; calculate arc length and sector area as fractions of the circle (Core: sector angles that are factors of 360°).
- 5.4Carry out calculations of the surface area and volume of cuboids, prisms and cylinders.
- 5.5Carry out calculations of perimeter and area of compound shapes, and volume and surface area of compound solids. (Extended for parts of solids such as hemispheres and frustums.)
Extended — Extended candidates only
- 5.3Calculate arc length and sector area for any angle, and distinguish minor and major sectors; give answers in terms of π where asked. (Extended)
- 5.4Carry out calculations of the surface area and volume of spheres, pyramids and right circular cones, including curved surface areas. (Extended)
In 30 seconds
- 1Area conversions square the length factor: 1 m² = 100 cm × 100 cm = 10 000 cm²; volume conversions cube it: 1 m³ = 1 000 000 cm³.
- 2Capacity: 1 litre = 1000 cm³ and 1 m³ = 1000 litres.
- 3Only the triangle area formula is printed on the formula sheet — know trapezium: A = ½(a + b)h (parallel sides a and b, perpendicular height h).
- 4A prism has the same cross-section all the way through: volume = area of cross-section × length.
- 5Sector fractions: arc = (θ⁄360) × 2πr and sector area = (θ⁄360) × πr² — circumference for length, circle area for area.
- 6Cone: curved surface area = πrl with l = slant height = √(r² + h²), and volume = ⅓πr²h.
- 7Sphere: surface area = 4πr² and volume = (4⁄3)πr³ — different coefficients, easy to swap by mistake.
- 8Round only at the end: keep π in the calculator (or at least 3.1416) until the final answer, then give 3 significant figures unless told otherwise.
Notes · 1 of 4
Units and conversions
Length, mass, area, volume and capacity — and the zeros that vanish.
Length conversions are factors of ten: 1 m = 100 cm and 1 cm = 10 mm. Area squares the factor: 1 m² = (100 cm)² = 10 000 cm², and 1 cm² = (10 mm)² = 100 mm². Volume cubes it: 1 m³ = (100 cm)³ = 1 000 000 cm³. For capacity, 1 litre = 1000 cm³ and 1 ml = 1 cm³, so a tank measured in centimetres converts straight to litres by dividing by 1000. Mass runs 1 kg = 1000 g and 1 tonne = 1000 kg.
The classic disaster is converting area with the length factor: writing 4.5 m² as 450 cm² loses two zeros — the truth is 45 000 cm². Sketch the conversion: metres to centimetres multiplies by 100, so square metres to square centimetres multiplies by 100 twice. The same logic gives 1 m³ = 100³ cm³, a factor of a million.
Notes · 2 of 4
Perimeter and area of plane shapes
Rectangles, triangles, parallelograms, trapeziums and compounds.
Perimeter is the distance around a shape; area is the covering inside. Rectangle: A = base × height. Triangle: A = ½bh (the one formula printed for you). Parallelogram: A = bh, using the perpendicular height, not the slanted side. Trapezium: A = ½(a + b)h, averaging the two parallel sides then multiplying by the perpendicular height. The formula sheet gives the triangle only, so these three come from memory.
Compound shapes split into pieces you know: an L-shaped floor is two rectangles; a running track is a rectangle plus semicircular ends. For a semicircle of diameter 50 m, the curved boundary is ½ × π × 50 and the area is ½ × π × 25². Add or subtract the parts, keeping π in the calculator until the end.
Notes · 3 of 4
Circles, arcs and sectors
Fractions of the circumference and of the area.
A full circle has circumference C = 2πr = πd and area A = πr². An arc is a fraction of the circumference and a sector is a fraction of the area — the same fraction in both cases: θ⁄360. So arc length = (θ⁄360) × 2πr and sector area = (θ⁄360) × πr². On Core papers the sector angle is always a friendly fraction of 360 (60°, 90°, 120° …); Extended papers use any angle, and you may be asked for the minor or major sector or the answer in terms of π.
Keep the two formulas straight by their shape: the arc formula starts from the circumference (a length), the sector formula from the area. If your arc length comes out bigger than the whole circumference, the fraction is wrong. “In terms of π” means do not press the π button at all: simplify the numbers and leave π as a symbol, e.g. 24π cm².
Notes · 4 of 4
Surface area and volume of solids
Prisms, cylinders, pyramids, cones, spheres and parts of solids.
Prisms keep one cross-section from end to end: volume = area of cross-section × length — a cuboid is a prism with a rectangular cross-section. A cylinder is a circular prism: volume πr²h, total surface area = two circles 2πr² plus the curved sheet 2πrh. A pyramid tapers to a point: volume = ⅓ × base area × height.
Extended. A cone is a circular pyramid: volume ⅓πr²h, curved surface area πrl, where l = √(r² + h²) is the slant height — note l and h are different lengths. A sphere has surface area 4πr² and volume (4⁄3)πr³; a hemisphere takes half of each volume but only half the curved surface plus one flat circle. Parts of solids (a cone with the top sliced off, a tank shaped like half a cylinder) are handled by subtracting or halving the full solids.
Revise
Key terms
The words examiners expect you to use precisely.
- Perimeter
- The total distance around the edge of a shape.A 9 m × 6 m garden needs 30 m of fence.
- Perpendicular height
- The height measured at right angles to the base.A = ½bh and A = ½(a + b)h both need it.
- Sector
- A pie-slice region bounded by two radii and an arc.A 90° sector is one quarter of the circle.
- Arc
- A part of the circumference between two radii.Arc = (θ⁄360) × 2πr.
- Prism
- A solid with a constant cross-section along its length.Volume = cross-section area × length.
- Slant height l (Extended)
- The distance from the edge of a cone’s base to its apex.l = √(r² + h²); πrl is the curved surface.
Show 2 more key termsShow fewer key terms
- Hemisphere (Extended)
- Half a sphere: half the volume, half the curved surface, plus one flat circle.Volume (2⁄3)πr³, curved SA 2πr².
- Part of a solid (Extended)
- A solid sliced by a plane, such as a frustum.Handle by subtracting or halving the full solid.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Converting area with the length factor: 4.5 m² = 450 cm².
Squaring shapes square the conversion factor.
Instead: 1 m² = 100² cm² = 10 000 cm², so 4.5 m² = 45 000 cm².
Mistake: Quoting cm³ as litres without dividing by 1000.
1 litre = 1000 cm³, not 1 cm³.
Instead: 60 000 cm³ = 60 litres.
Show 10 more mistakesShow fewer mistakes
Mistake: Using the slanted side as the height of a parallelogram or trapezium.
The area formula needs the perpendicular height.
Instead: Drop a right angle from the top edge to the base before measuring h.
Mistake: Substituting the diameter into πr².
A = πr² needs the radius; with d = 10, r = 5.
Instead: Halve the diameter first: A = π(5²) = 25π.
Mistake: Computing arc length with the area formula.
Lengths come from the circumference, not from πr².
Instead: Arc = (θ⁄360) × 2πr.
Mistake: Using the height h instead of the slant l in πrl.
The curved surface unrolls along the slant, not the vertical axis.
Instead: Find l = √(r² + h²) first: r = 6, h = 8 gives l = 10.
Mistake: Forgetting the circular ends of a cylinder’s surface area.
2πrh is only the curved sheet.
Instead: Total SA = 2πr² + 2πrh.
Mistake: Swapping the sphere coefficients: SA = (4⁄3)πr³.
The area formula is 4πr²; the volume is (4⁄3)πr³.
Instead: Area has r², volume has r³ — match the power to the dimension.
Mistake: Rounding π to 3.14 at the start of a calculation.
Early rounding pollutes every later step.
Instead: Keep the calculator’s π (or 3.14159…) until the final answer, then round to 3 s.f.
Mistake: Treating 1 cm³ as 1 litre.
A litre is a thousand cubic centimetres.
Instead: 1 cm³ = 1 ml, so 1 000 cm³ = 1 litre; 500 000 cm³ is 500 litres, not 500 000.
Mistake: Halving the slant height to get a cone’s perpendicular height.
The slant is the hypotenuse of the right triangle, not twice the height.
Instead: l² = r² + h²: radius 6 with slant 10 gives height 8.
Mistake: Multiplying three lengths for an area (or two for a volume).
The units give it away: m² needs two lengths, m³ needs three.
Instead: Match the calculation to the unit: 6 × 4 = 24 m², then × 3 = 72 m³.
Check yourself
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 15 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Topic 15: metric unit conversions including area units and capacity, perimeter and area of rectangles, triangles, parallelograms and trapeziums, circles, arcs and sectors, volumes and surface areas of prisms, cylinders, cones, spheres and compound solids.
Mark scheme
PROTopic 15 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
Which area formulas are on the IGCSE Maths formula sheet?
The sheet gives the area of a **triangle** (½bh), the area and circumference of a circle (A = πr², C = 2πr), and the volume/surface formulas for the harder solids. The areas of the rectangle, parallelogram and trapezium are **not** printed — learn A = bh and A = ½(a + b)h, and know that the trapezium formula is just the average of the parallel sides times the perpendicular height.
How do I stop mixing up arc length and sector area?
Both use the same fraction θ⁄360, but they start from different wholes: arc length is the fraction of the **circumference** (2πr — a length), sector area is the fraction of the **area** (πr² — an area). A quick unit check catches most slips: an arc is measured in cm, a sector in cm².
How accurately should I give mensuration answers?
Keep π inside the calculator (or at least 3.1416) until the very end, then round to **3 significant figures** unless the question says otherwise. If the question asks for the answer “in terms of π”, do not evaluate π at all — simplify the numbers and leave the π symbol, e.g. 24π cm².
What is the fastest way to remember the volume and capacity link?
1 cm³ = 1 millilitre, so 1000 cm³ = 1 litre, and 1 m³ = 100 × 100 × 100 = 1 000 000 cm³ = 1000 litres. Measure a tank in centimetres, multiply for the cm³, divide by 1000 for litres — that chain answers almost every capacity question.
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No. Cambridge IGCSE and Cambridge are trademarks of Cambridge University Press & Assessment. QuizQuip is independent — not affiliated with, endorsed by, or connected to Cambridge. All our notes and questions are original material written against the published 0580 syllabus; we do not host Cambridge past papers.
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Topic 16: Trigonometry