LEARNING OBJECTIVES
By the end of this chapter, you should be able to:
- convert metric units of length, area, volume, capacity and mass
- calculate perimeter and area of common shapes
- work with circles, arcs and sectors
- calculate surface areas and volumes of solids
- solve compound-shape and compound-solid problems
THE BIG IDEA
Measure length, area, capacity, surface area and volume with correct units and efficient decomposition.
Mensuration applies geometry to measurement. A formula gives a numerical result, but units explain what was measured.
Compound figures become manageable when they are split into familiar parts. Draw or label those parts before calculating.
SECTION 01
Units and conversions
A linear conversion factor is squared for area and cubed for volume. For example, 1 m = 100 cm, so 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³.
Capacity connects to volume: 1 cm³ = 1 ml and 1000 cm³ = 1 litre.
KEY IDEAS
- Convert every measurement to compatible units before using a formula.
- Mass uses g and kg; capacity uses ml and l.
Converting 0.036 m³ to litres
- 1 m³ = 1 000 000 cm³.
- 0.036 m³ = 36 000 cm³.
- 1000 cm³ = 1 litre.
Answer: 0.036 m³ = 36 litres.
SECTION 02
Perimeter and area
Perimeter is the total boundary length. Area measures the surface enclosed.
A parallelogram and triangle use perpendicular height, not a sloping side. A trapezium's parallel sides are the pair used in its area formula.
Area of a trapezium
- Parallel sides are 8.2 cm and 13.6 cm; perpendicular height is 5 cm.
- Area = 1/2(8.2 + 13.6) × 5.
- = 1/2 × 21.8 × 5.
Answer: Area = 54.5 cm².
SECTION 03
Circles, arcs and sectors
A sector is a fraction of a full circle. Use angle/360 to take the same fraction of circumference or area.
Keep π in the answer when an exact form is requested; otherwise use the calculator value and round only at the end.
Major sector with radius 9 cm and minor angle 80°
- Major angle = 360° − 80° = 280°.
- Major sector area = 280/360 × π × 9².
- Simplify 280/360 to 7/9.
Answer: Area = 63π cm², approximately 197.9 cm².
SECTION 04
Surface area and volume
Volume measures space inside a solid. Surface area totals all exposed faces, including circular ends where present.
A prism has volume equal to cross-sectional area times length. Pyramids and cones use one third of base area times perpendicular height.
Volume of a triangular prism
- Triangle base = 7 cm and perpendicular height = 4 cm.
- Cross-sectional area = 1/2 × 7 × 4 = 14 cm².
- Prism length = 12 cm.
- Volume = 14 × 12.
Answer: Volume = 168 cm³.
SECTION 05
Compound shapes and solids
Split a compound shape into non-overlapping familiar regions, then add or subtract. For perimeter, count only the outside boundary.
For compound solids, internal joining faces are not exposed and must not be included in surface area.
A frustum can be treated as a large cone or pyramid with a smaller similar one removed.
Volume of a conical frustum
- A large cone has volume 360π cm³.
- The removed similar cone has linear scale factor 1/2.
- Its volume scale factor is (1/2)³ = 1/8, so its volume is 45π cm³.
- Subtract the removed volume.
Answer: Frustum volume = 315π cm³.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Square or cube linear conversion factors for area or volume.
- Perimeter measures boundary; area measures surface.
- Arc and sector calculations use θ/360.
- A prism uses uniform cross-section area × length.
- Surface area includes only exposed faces.
- Compound figures are solved by careful addition or subtraction.