Sec 3 E-Math: Mensuration

Overview

What you’ll learn

Calculate the areas of parallelograms and trapeziums.
Find the perimeter and area of composite plane figures built from rectangles, triangles and circles.
Calculate arc length, sector area and the area of a segment of a circle.
Use radian measure and convert between radians and degrees, including s = rθ and A = ½r²θ.
Calculate the volume and surface area of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres.
Solve problems involving composite solids, using conversions between cm² and m² and between cm³ and m³.

In 30 seconds

1Trapezium area = ½(a + b)h where a and b are the parallel sides; parallelogram area = base × height.
2Split composite shapes into rectangles, triangles and circle pieces — and keep perimeters to the outer boundary only.
3With degrees: arc = (θ/360) × 2πr and sector = (θ/360) × πr². With radians: arc s = rθ and sector A = ½r²θ.
4Segment area = sector area − the triangle made by the two radii.
5π radians = 180°, so 1 rad ≈ 57.3°; convert with × π/180 or × 180/π.
6Prism volume = area of cross-section × length; cylinder = πr²h.
7Cone = ⅓πr²h with slant l = √(r² + h²); pyramid = ⅓ × base area × height; sphere V = (4/3)πr³, S = 4πr².
81 m² 10 000 cm² and 1 m³ 1 000 000 cm³ — square and cube the 100, never just 100 itself.

Notes · 1 of 5

Areas of plane figures

The standard formulas, then composites by splitting and adding.

Core area formulas
Shape	Formula
Rectangle	length × width
Triangle	½ × base × height
Parallelogram	base × height
Trapezium	½(a + b)h, a and b the parallel sides
Circle	πr²

A composite figure mixes shapes. Two habits pay for themselves: split along visible lines into standard shapes and add; and for a perimeter, walk the outer boundary only — internal split lines never count. Curved boundaries use the arc formulas from the next section.

Heights must be perpendicular to bases: a slanted side of a parallelogram is not its height. When a diagram gives a slant and a height, use the height for areas and the slant only for perimeters.

Notes · 2 of 5

Arcs, sectors and segments

Circumference and area, proportioned by the angle.

An arc is a piece of the circumference; a sector is the pie slice it bounds with two radii; a segment is the region between a chord and its arc. With the angle θ in degrees, the circle’s totals are shared out in proportion θ/360: arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².

1Arc
r = 6 cm, θ = 50°: arc = (50/360) × 2π(6) = 5.24 cm.
2Sector
Same sector: area = (50/360) × π(36) = 15.7 cm².
3Segment
For a 90° sector of radius 10: area = ¼π(100) − ½(10)(10) = 78.5 − 50 = 28.5 cm² — sector minus the right triangle of the two radii and the chord.

Notes · 3 of 5

Radian measure

The natural angle unit: arc = radius makes 1 radian.

A radian is the angle standing on an arc exactly as long as the radius. A full turn is 2π radians, so π radians = 180° and 1 radian ≈ 57.3°. Convert degrees to radians by multiplying by π/180, and back by 180/π: 60° = π/3 ≈ 1.05 rad; 2.5 rad = 2.5 × (180/π) ≈ 143.2°.

Degree–radian reference
Degrees	30°	45°	60°	90°	180°	360°
Radians	π/6	π/4	π/3	π/2	π	2π

Radians simplify the circle formulas beautifully: arc length s = rθ and sector area A = ½r²θ, with θ in radians. For r = 5 and θ = 0.8: s = 4 cm and A = ½(25)(0.8) = 10 cm². Always confirm the angle is in radians before using these two — the degree versions have the θ/360 factor instead.

Notes · 4 of 5

Volumes and surface areas of solids

One prism idea, one pyramid idea, and three curved-surface classics.

Solid formulas
Solid	Volume	Surface area
Cuboid	lwh	2(lw + lh + wh)
Prism	cross-section area × length	sum of all faces
Cylinder	πr²h	2πr² + 2πrh
Pyramid	⅓ × base area × height	base + triangular faces
Cone	⅓πr²h	πr² + πrl (l = √(r² + h²))
Sphere	(4/3)πr³	4πr²

A prism is any solid with the same cross-section all the way through — its volume is cross-section area × length. A pyramid or cone tapers to a point: exactly one third of the matching prism or cylinder. The cone’s slant height l comes from Pythagoras, l = √(r² + h²), and its curved surface unrolls into a sector of area πrl.

Notes · 5 of 5

Composite solids and unit conversions

Adding solids, subtracting cavities, and converting areas and volumes honestly.

A composite solid joins or hollows out standard solids. Add volumes for joined pieces; subtract for drilled holes and cavities. Surface area needs a boundary walk in 3D: faces hidden where two solids meet drop out, and cut faces appear. Sketch the solid, list its faces, then total them.

Conversions: 1 m = 100 cm ⇒ 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³.
A tank 1.2 m × 80 cm × 50 cm has volume 1.2 × 0.8 × 0.5 = 0.48 m³ = 480 000 cm³ = 480 litres.
Convert before adding: a question mixing metres and centimetres is testing exactly this.

Real-world answers should carry sensible units and rounding: capacity in litres (1 litre = 1000 cm³), paint in m², and 3 significant figures unless the question says otherwise.

Revise

Key terms

The words examiners expect you to use precisely.

Trapezium area: ½(a + b)h, with a and b the parallel sides and h the perpendicular height between them.
Composite figure: A plane shape built from standard shapes; split to find area, walk the boundary for perimeter.
Arc length: (θ/360) × 2πr in degrees, or s = rθ in radians.
Sector: The region between two radii and an arc; area (θ/360)πr² or ½r²θ.
Segment: The region between a chord and its arc; area = sector − triangle.
Radian: The angle subtending an arc equal to the radius; π rad = 180°.

Show 5 more key terms

Prism: A solid with constant cross-section; volume = cross-section area × length.
Slant height: A cone’s l = √(r² + h²), used in πrl for the curved surface.
Sphere: V = (4/3)πr³ and surface area 4πr².
Composite solid: A solid built from standard solids; add volumes, and count only exposed faces for surface area.
Litre: 1000 cm³; used for capacity answers.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Using the slanted side of a parallelogram as the height.
Area needs the perpendicular height.
Instead: base × perpendicular height; the slant only matters for perimeters.
Mistake: Adding internal split lines into a composite perimeter.
The perimeter is the outer boundary only.
Instead: Walk around the outside; the rectangle–semicircle join line stays out.

Show 10 more mistakes

Mistake: Using arc = (θ/360) × πr² (the sector formula) for arc length.
Length is one-dimensional: it comes from the circumference 2πr.
Instead: arc = (θ/360) × 2πr; area = (θ/360) × πr².
Mistake: Using s = rθ with θ still in degrees.
The radian formulas need radians.
Instead: Convert first (×π/180) or use the degree versions with θ/360.
Mistake: Reporting the segment area as the full sector area.
The chord cuts the triangle away.
Instead: segment = sector − triangle (the triangle has sides r, r and included angle θ).
Mistake: Converting 1 m² as 100 cm² (or 1 m³ as 100 cm³).
Each dimension scales by 100, so area squares and volume cubes the factor.
Instead: 1 m² = 10 000 cm²; 1 m³ = 1 000 000 cm³.
Mistake: Forgetting the ⅓ in cone and pyramid volumes.
A cone is exactly one third of its matching cylinder.
Instead: Cone ⅓πr²h; pyramid ⅓ × base × height.
Mistake: Using the vertical height where the slant height belongs (or vice versa) in a cone.
πrl needs the slant; ⅓πr²h needs the vertical.
Instead: l = √(r² + h²) converts between them.
Mistake: Counting the hidden joining face in a composite solid’s surface area.
Faces glued together are interior.
Instead: Surface area = exposed faces only.
Mistake: Halving a sphere formula as if it were a hemisphere by halving volume but quartering… anything.
A hemisphere is half the volume and half the curved surface, plus a new flat circular face.
Instead: Hemisphere: V = (2/3)πr³; surface = 2πr² + πr² = 3πr² (including the flat face).
Mistake: Mixing units inside one calculation: adding 1.2 m to 80 cm.
The sum is meaningless until units agree.
Instead: Convert everything first (80 cm = 0.8 m) and state the answer’s unit.
Mistake: Rounding π to 3.14 (or 22/7) when the question expects calculator accuracy.
Premature rounding shifts the third significant figure.
Instead: Use the calculator π throughout and round only the final answer.

Check yourself

Quick quiz

32 questions with instant explanations. About 5 minutes.

Question 1 of 3

Question 1: current question
Question 2: not answered
Question 3: not answered

Find areas of plane figures

A trapezium has parallel sides 8 cm and 12 cm with height 5 cm. Its area is

Practise

Topical questions

Exam-style practice in three tiers, from recall to data questions.

7Foundation
7Standard
6Challenging

20 questions worth 86 marks, each with a model answer, mark scheme and examiner tips.

Practise topical questions

Keep practising

Practise this chapter

Jump straight to the questions for this chapter, the chapters either side of it, and this year’s assessment papers.

Previous chapterTrigonometry: topical questions This chapterTopical questions20 questions with answers

Test yourself

Full papers for this subject. Each school sets its own scope — check your assessment outline.

Exam practice

Practice paper

A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.

Question paper

Chapter 13 Practice Paper

30 marks · 40 min · 4 pages

7 multiple-choice questions and structured questions with answer lines. Chapter 13 (G5 Mensuration: composite plane figures, arc/sector/segment, radians, volumes and surface areas of solids, composite solids, unit conversions).

Download PDF Print view

Mark scheme

PRO

Chapter 13 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.

Print view

Where this chapter is tested

Each school sets its own scope — check your assessment outline.

Questions

Frequently asked questions

What does Sec 3 E-Math Chapter 13 cover?

It covers syllabus sub-topic G5 of O-Level 4052 / SEC G3 K310 in full: parallelogram and trapezium areas, composite plane figures, arc length / sector area / segment area, radian measure and conversions, volumes and surface areas of all the standard solids, composite solids, and cm²–m² / cm³–m³ conversions.

When do I use radians instead of degrees?

Use whichever the question gives, converting as needed. In radian mode the formulas shrink to arc = rθ and sector = ½r²θ; in degrees they carry a θ/360 factor. Check your calculator’s angle mode before any sin/cos/tan work.

How do I handle surface area of composite solids?

List the exposed faces only. Where two solids meet, the shared circle or rectangle is hidden and must be excluded — while any newly cut face (like the flat top of a hemisphere hole) must be included. A labelled sketch of the faces prevents double counting.

What is the most common unit-conversion error?

Treating 1 m² as 100 cm². Since both dimensions scale by 100, area scales by 10 000 and volume by 1 000 000. Convert lengths first, then compute — or square/cube the conversion factor deliberately.

When is this chapter usually tested in school?

Mensuration is a guaranteed End-of-Year topic: usually one composite plane figure, one arc/sector/segment or radian item, and a solid volume–surface question worth 6–10 marks. Formula sheets help, but the marks sit in correct substitution and honest unit conversions.

Sec 3 E-Math Mensuration notes