Cambridge O Level Mathematics: Mensuration (4024) – Revision Notes

These revision notes are aligned to the Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for examination in 2025, 2026 and 2027, topic 5 Mensuration, sections 5.1 to 5.5. The syllabus is not tiered. For full explanations and longer worked examples, read the study guide. For timed practice, use the mensuration practice set.

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5.1 Units: convert fast

Quantity	Facts to know
Length	1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m
Area	multiply the length factor by itself: 1 m² = 10 000 cm², 1 cm² = 100 mm², 1 km² = 1 000 000 m²
Volume	cube the length factor: 1 m³ = 1 000 000 cm³
Capacity	1 cm³ = 1 ml, 1 l = 1000 cm³, 1 m³ = 1000 l
Mass	1 kg = 1000 g

Smaller unit means a bigger number. Larger unit means a smaller number.

Reminder: 0.06 m² = 0.06 × 10 000 = 600 cm². And 2.5 m³ = 2500 litres.

5.2 to 5.4 Formulas

The syllabus says a list of formulas is included on page 2 of the examination papers. The items it names for this topic are marked “given”. Everything else you must know.

Result	Formula	Status
Rectangle area	lw	learn
Triangle area	½bh	given
Parallelogram area	bh	learn
Trapezium area	½(a + b)h	learn
Circle area	πr²	given
Circumference	2πr	given
Arc length	(θ/360) × 2πr	learn (fraction of circumference)
Sector area	(θ/360) × πr²	learn (fraction of area)
Prism volume	Al (cross-section area times length)	given
Cylinder volume	πr²h	given
Cylinder curved surface	2πrh	given
Cone volume	⅓πr²h	given
Cone curved surface	πrl	given
Sphere volume	(4/3)πr³	given
Sphere surface area	4πr²	given
Pyramid volume	⅓ × base area × h	given
Cuboid surface area	2(lw + lh + wh)	learn

Cuboid volume is lwh, which you build from the prism rule.

Method boxes

Perimeter and area of a sector

1. Write the fraction θ/360 and simplify it.
2. Arc = fraction × 2πr. Area = fraction × πr².
3. Perimeter of the sector = arc + 2r (two straight radii).
4. If the question says “in terms of π”, keep π in the answer.

Surface area of a solid

1. List every face: flat faces, curved faces, ends.
2. Decide if any face is missing (open top) or hidden (a join).
3. Find each area and add.
4. For a cone or pyramid, get l or the slant height from Pythagoras first.

Volume of a compound solid

1. Split into cones, cylinders, hemispheres, prisms or pyramids.
2. Work out any missing height by subtraction.
3. Add volumes. If a piece is removed, subtract.

Shaded regions

1. Area of the large shape minus area of the small shape.
2. Keep π in until the last line.

Worked reminders

Ring (annulus). Outer radius 10 cm, inner radius 6 cm. Area = π(10² − 6²) = π × 64 = 64π cm².

Quarter circle. Radius 10 cm. Area = ¼ × π × 100 = 25π cm². Perimeter = 10 + 10 + ¼ × 2π × 10 = 20 + 5π cm.

Solid hemisphere, radius 3 cm. Volume = ⅔ × π × 27 = 18π cm³. Curved surface = 2π × 9 = 18π. Total surface area = 18π + π × 9 = 27π cm² because the flat face counts when the solid is closed.

Compound solid. A cylinder of radius 4 cm and height 9 cm has a hemisphere of the same radius fixed on top. Volume = π × 16 × 9 + ⅔ × π × 64 = 144π + (128/3)π = (560/3)π cm³. The curved surface is 2π × 4 × 9 + 2π × 16 = 72π + 32π, plus the base 16π, giving 120π cm² in total. The circle where the two parts meet is a join, so it is left out.

Mixed units. A tank 1.2 m long, 50 cm wide and 40 cm deep: change to one unit first. 120 × 50 × 40 = 240 000 cm³ = 240 litres.

Recasting. When one solid is melted and made into another, the volume stays the same. Equate the two volumes and solve for the missing length.

Must-know distinctions

Idea	Careful about
Height v slant height	Area and volume use the perpendicular height. Curved surface of a cone uses the sloping edge l.
Radius v diameter	Halve the diameter before using any circle formula.
Perimeter v area	Perimeter is a length (cm). Area is cm². Volume is cm³.
Curved v total surface	Total includes flat ends. Open containers leave off a lid.
Minor v major sector	Minor sector has angle below 180°, major above 180°. The same formulas work.
In terms of π v decimal	Leave π in if asked. Otherwise give 3 significant figures.
Capacity v volume	Capacity is measured in ml or litres, volume in cm³ or m³.

The syllabus says to give non-exact answers to 3 significant figures unless a different accuracy is specified, and to avoid rounding before the final answer. For a calculator paper, use π from your calculator or 3.142.

Parts of shapes and solids

• A half or quarter of a circle, cylinder or sphere is half or a quarter of the whole.
• A cut adds new flat faces. A half-cylinder has a flat rectangle as well as its two semicircular ends.
• A frustum is a cone with a smaller, similar cone removed from the top. Find both cones, then subtract volumes. Add the base, the top circle and the difference of the two curved surfaces.
• A prism can have any uniform cross-section, so find the cross-section area first.

Quick self-test

Try these without notes. Questions 3 and 10 are best done without a calculator.

1. Convert 0.06 m² to cm².
2. Convert 4800 cm³ to litres.
3. A trapezium has parallel sides 9 cm and 15 cm and height 8 cm. Find its area.
4. A parallelogram has area 72 cm² and base 9 cm. Find its perpendicular height.
5. A sector has radius 8 cm and angle 135°. Find its arc length and area in terms of π.
6. A sector of radius 8 cm has area 40π cm². Find its angle.
7. A sphere has radius 6 cm. Find its volume and surface area in terms of π.
8. A cone has base radius 5 cm and sloping edge 13 cm. Find its perpendicular height, volume and curved surface area, in terms of π.
9. A cylinder with base radius 6 cm has volume 360π cm³. Find its height.
10. A square-based pyramid has base edge 6 cm and vertical height 4 cm. Find its volume and total surface area.
11. A cuboid measures 8 cm by 5 cm by 3 cm. Find its total surface area.
12. A sphere of radius 3 cm is melted and recast as a cone with base radius 3 cm. Find the cone’s height.

Answers

1. 0.06 × 10 000 = 600 cm².
2. 4800 ÷ 1000 = 4.8 litres.
3. ½ × (9 + 15) × 8 = 96 cm².
4. 72 ÷ 9 = 8 cm.
5. Fraction = 135/360 = 3/8. Arc = 3/8 × 16π = 6π cm. Area = 3/8 × 64π = 24π cm².
6. 40π = (θ/360) × 64π, so θ/360 = 5/8, and θ = 225°.
7. Volume = (4/3) × π × 216 = 288π cm³. Surface area = 4 × π × 36 = 144π cm².
8. h² = 13² − 5² = 144, so h = 12 cm. Volume = ⅓ × π × 25 × 12 = 100π cm³. Curved surface = π × 5 × 13 = 65π cm².
9. π × 36 × h = 360π, so h = 10 cm.
10. Volume = ⅓ × 36 × 4 = 48 cm³. Slant height = √(4² + 3²) = 5 cm. Four faces = 4 × ½ × 6 × 5 = 60. Total = 36 + 60 = 96 cm².
11. 2 × (40 + 24 + 15) = 158 cm².
12. Sphere volume = (4/3) × π × 27 = 36π. Cone: ⅓ × π × 9 × h = 36π, so 3πh = 36π and h = 12 cm.

Where marks are usually lost

• Converting between area units with the length factor instead of its square.
• Using the slant side of a trapezium or parallelogram as the height.
• Using the diameter as the radius, especially when a question gives the width of a circular shape.
• Forgetting the radii in the perimeter of a sector.
• Taking a major sector as the same angle as the minor one, instead of 360° minus it.
• Using the vertical height of a cone where the curved surface needs l, or the reverse.
• Including a lid on an open cup, or missing the flat face on a solid hemisphere.
• Counting a join between two pieces of a compound solid in the surface area.
• Rounding early, then giving a final answer that is not accurate to 3 significant figures.
• Writing cm² where cm³ is needed, or leaving units off.

Before moving on, work through the mensuration practice set and revisit the study guide for any weak section.

Frequently asked questions

Which mensuration formulas are given in the Cambridge O Level Mathematics 4024 exam? According to these notes, formulas such as the area of a circle, circumference, cylinder, cone, sphere, prism and pyramid volumes are given. You must learn the rectangle, parallelogram and trapezium areas, arc length, sector area and cuboid surface area.

How do you find arc length and sector area? Use the fraction of the circle: arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². The same formulas work for major sectors, where the angle is above 180°.

How do you find the volume of a frustum? A frustum is a cone with a smaller, similar cone removed from the top. Find the volumes of both cones and subtract.

What happens to the volume when a solid is melted and recast? The volume stays the same. Equate the volumes of the two solids and solve for the missing length.

What is the difference between capacity and volume? Capacity is measured in ml or litres, while volume is measured in cm³ or m³. 1 cm³ = 1 ml and 1 litre = 1000 cm³.

Official syllabus

Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for examination in 2025, 2026 and 2027, Cambridge International Education, topic 5 Mensuration.