Overview
What you’ll learn
- Convert between metric units of length, area and volume.
- Find perimeters and areas of rectangles, triangles, parallelograms and trapeziums.
- Find areas of composite plane figures by splitting and subtracting.
- Find volumes and surface areas of cubes, cuboids, prisms and cylinders.
- Solve real-world problems involving measurement, including costs per square metre and filling boxes.
In 30 seconds
- 1Area of triangle = ½ × base × height, with the height perpendicular to the base.
- 2Area of trapezium = ½ (a + b) × h — average the parallel sides, times the height.
- 3Volume of a prism = cross-section area × length — the formula that covers cubes, cuboids and cylinders alike.
- 4Surface area = the total of every face; draw the net so no face is doubled or forgotten.
- 51 m² = 100 cm × 100 cm = 10 000 cm²; 1 cm³ = 1 mL.
- 6Composite figures: split into known shapes and add, or subtract the hole from the whole.
Notes · 1 of 5
Converting units of length, area and volume
Lengths scale by 10, areas by 100, volumes by 1 000 — per step.
| Quantity | Conversion | Why |
|---|---|---|
| 1 m = 100 cm | length | one step of 100 (centi) |
| 1 m² = 10 000 cm² | area | 100 × 100: two length steps |
| 1 m³ = 1 000 000 cm³ | volume | 100 × 100 × 100: three length steps |
| 1 L = 1 000 mL = 1 000 cm³ | capacity | a 10 cm cube holds exactly 1 L |
| 1 ha = 10 000 m² | land area | a 100 m × 100 m square |
Notes · 2 of 5
Perimeter and area of plane figures
Each formula measures a different idea: distance around, or coverage inside.
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle l × b | 2(l + b) | l × b |
| Triangle | sum of the three sides | ½ × base × height |
| Parallelogram | 2(a + b) | base × height |
| Trapezium | sum of the four sides | ½ (a + b) × h |
The height in these formulas is always the perpendicular distance between the base and the tip (or the far side) — never the slanted edge. A triangle with base 14 cm and height 6 cm has area ½ × 14 × 6 = 42 cm² even if its other two sides are long and slanted.
A parallelogram is a pushed-over rectangle: sliding the triangle from one end to the other shows why its area is still base × height. A trapezium is two triangles glued along the height, which is why the two parallel sides are averaged: ½ (a + b) × h.
Notes · 3 of 5
Volume of prisms and cylinders
One idea covers every prism: slide a cross-section along a length.
A prism has the same cross-section all the way through its length. Its volume is cross-section area × length: a cuboid is a rectangular prism (lb × h), a cube is a square prism (x² × x = x³), and a cylinder is a circular prism (pi × r² × h).
| Solid | Cross-section | Volume |
|---|---|---|
| Cuboid 6 × 4 × 2.5 | rectangle 6 × 4 | 24 × 2.5 = 60 cm³ |
| Cube of edge 3 | square 3 × 3 | 9 × 3 = 27 cm³ |
| Cylinder r = 7, h = 12 (pi = 22/7) | circle: (22/7) × 49 | 154 × 12 = 1 848 cm³ |
Notes · 4 of 5
Surface area of prisms and cylinders
Unfold the solid into its net, then add up every face once.
The surface area of a cuboid is the total of its three pairs of faces: 2(lb + lh + bh). For a 5 × 4 × 3 cm cuboid: 2 × (20 + 15 + 12) = 94 cm². Drawing the net of six rectangles keeps the pairs honest — the most common slip is counting one face twice and forgetting another.
| Part | Shape when unfolded | Formula |
|---|---|---|
| Two circular ends | two circles | 2 × pi × r² |
| Curved surface | a rectangle: 2 × pi × r wide, h tall | 2 × pi × r × h |
| Total (closed cylinder) | all of the above | 2 × pi × r × (r + h) |
The curved surface unrolls into a rectangle whose width is the circumference (2 × pi × r) and whose height is the cylinder’s height — that is exactly why the formula works. A tin of radius 7 cm and height 15 cm has curved surface 2 × (22/7) × 7 × 15 = 660 cm².
Notes · 5 of 5
Measurement in real contexts
Rates bolt onto measurements: dollars per square metre, litres per minute, tiles per floor.
Exam questions rarely stop at the measurement. They attach a rate — a cost per square metre, a flow per minute, a number of tiles — and expect you to chain the answer into the real world. The pattern is always the same: measure first, then multiply or divide by the rate with matching units.
| Question type | Measure first | Then |
|---|---|---|
| Turf costs $12 per m² | area in m² | multiply: m² × $/m² = $ |
| Tank empties at 20 L/min | volume in litres | divide: L ÷ (L/min) = min |
| Tiles of 50 cm sides | floor area and tile area in the same unit | divide the areas (check the tiles actually fit) |
Revise
Key terms
The words examiners expect you to use precisely.
- Perimeter
- The total distance around the edge of a flat shape.
- Area
- The amount of surface a flat shape covers, measured in square units.
- Perpendicular height
- The straight-line distance from a base to the opposite point or side, at 90° to the base.
- Trapezium
- A quadrilateral with exactly one pair of parallel sides.
- Parallelogram
- A quadrilateral with two pairs of parallel sides; its area is base × height.
- Composite figure
- A shape built from simpler figures; find its area by splitting or subtracting.
Show 6 more key termsShow fewer key terms
- Prism
- A solid with the same cross-section along its whole length; volume = cross-section area × length.
- Cylinder
- A circular prism; volume = pi × r² × h.
- Surface area
- The total area of all the faces of a solid, found from its net.
- Net
- The flat pattern that folds into a solid.
- Circumference
- The perimeter of a circle: 2 × pi × r.
- pi
- The ratio of a circle’s circumference to its diameter, about 3.14159; often taken as 22/7 in school problems.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Using the slanted side as the height of a triangle or parallelogram.
The formulas need the perpendicular distance between the base and the tip, not the length of a leaning edge.
Instead: Drop a perpendicular from the tip to the base line; that length is the height.
Mistake: Converting area with the length factor: 12 m² changed to 1 200 cm².
Area scales with the square of the length factor: 1 m² = 100 × 100 = 10 000 cm².
Instead: 12 m² = 12 × 10 000 = 120 000 cm².
Show 8 more mistakesShow fewer mistakes
Mistake: Answering a perimeter question with the area formula, e.g. 2(l + b) quoted in cm².
Perimeter is a distance around the edge, measured in plain length units.
Instead: Perimeter in cm or m; area in cm² or m². Check the unit matches the question.
Mistake: Computing cylinder volume as 2 × pi × r × h.
That is the curved surface formula; volume needs the circular cross-section area first.
Instead: Volume = pi × r² × h; curved surface = 2 × pi × r × h.
Mistake: Adding only three faces for the surface area of a cuboid.
A cuboid has six faces in three identical pairs; three faces is half the surface.
Instead: Total = 2(lb + lh + bh), or add all six from the net.
Mistake: Using a volume ratio for packing without checking the shapes fit.
A volume ratio assumes the blocks tile the box with no awkward gaps.
Instead: Verify the arrangement along each dimension (here 2 × 3 × 4 works perfectly).
Mistake: Mixing pi forms mid-solution, e.g. starting with 22/7 and finishing with a calculator pi button.
The two values differ slightly, so answers disagree in the last digits.
Instead: Follow the paper’s instruction (usually “take pi as 22/7”) from the first line to the last.
Mistake: Converting volume with the length factor once, e.g. claiming 1 m³ = 1 000 cm³.
Volume scales with the cube of the length factor: 100 × 100 × 100.
Instead: 1 m³ = 1 000 000 cm³ (and 1 000 L, since 1 000 cm³ = 1 L).
Mistake: Using the diameter as the radius, e.g. computing a tin of diameter 14 cm with r = 14.
Every circle formula takes the radius, which is half the diameter.
Instead: Halve first: d = 14 cm gives r = 7 cm, then use pi × r² × h.
Mistake: Assuming a rectangular notch cut from a corner changes the perimeter.
The notch’s two inner edges exactly replace the two outer edges that were removed, so the total edge length is unchanged.
Instead: Cutting a corner notch leaves the perimeter the same while reducing the area.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Find the area of a rectangle
A rectangle measures 8 cm by 5 cm. What is its area?
Ready for more practice on Chapter 9: Perimeter, Area, Volume and Surface Area?
Generate a fresh 5-question drill built from this chapter’s key points. Free with your study account.
Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 6Foundation
- 5Standard
- 4Challenging
15 questions worth 54 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam 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 9 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 9.
Mark scheme
PROChapter 9 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.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 3Coming soonWeighted Assessment 3 (WA3)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 1 Math Chapter 9 about?
Chapter 9 covers measurement: converting metric units of length, area and volume, perimeters and areas of plane figures including trapeziums and composite shapes, and volumes and surface areas of cubes, cuboids, prisms and cylinders.
How many cm² are in a m²?
10 000. The length factor is 100, and area squares it: 1 m² = 100 cm × 100 cm. Likewise 1 m³ = 1 000 000 cm³ because the factor is used three times.
What is the formula for the volume of a prism?
Volume = area of cross-section × length. A cuboid is a rectangular prism, and a cylinder is a circular prism with volume pi × r² × h — the same idea with a circle as the cross-section.
How do I find the surface area of a cylinder?
Unroll it. The curved surface becomes a rectangle of width 2 × pi × r (the circumference) and height h, so the curved area is 2 × pi × r × h. Add the two circular ends, 2 × pi × r², for a closed cylinder.
When is this chapter tested?
Measurement questions usually appear in Weighted Assessment 3 and the end-of-year exam, often as word problems with costs or packing. Every school sets its own scope, so check your school’s assessment outline.
Up next
Chapter 10: Statistical Data Handling