Measurement – IB / IGCSE Mathematics
(Study Notes – Exam‑oriented, with depth, derivations, key formulae and a hard practice problem)
1. Introduction
- “Measurement” is a core theme in both IGCSE and IB (SL/HL).
- It tests conceptual understanding, unit conversion, application of formulas, and error analysis.
- Exam style: Multiple‑choice, short answer, extended response, and calculation questions.
- Key skills:
- Converting between metric and imperial units.
- Calculating perimeter, area, surface area, and volume of standard shapes.
- Approximating the area of irregular shapes.
- Using the distance formula and slope in the coordinate plane.
- Converting between degrees and radians; measuring angles.
- Understanding significant figures and measurement error.
2. Units & Conversions
| Quantity | SI base unit | Common Imperial unit | Conversion factor |
|---|---|---|---|
| Length | metre (m) | foot (ft) | 1 m ≈ 3.28084 ft |
| centimetre (cm) | inch (in) | 1 cm = 0.393701 in | |
| kilometre (km) | mile (mi) | 1 km = 0.621371 mi | |
| Area | square metre (m²) | square foot (ft²) | 1 m² ≈ 10.7639 ft² |
| square centimetre (cm²) | square inch (in²) | 1 cm² = 0.155 in² | |
| hectare (ha) | acre (ac) | 1 ha = 2.47105 ac | |
| Volume | cubic metre (m³) | cubic foot (ft³) | 1 m³ ≈ 35.3147 ft³ |
| litre (L) | gallon (gal) | 1 L = 0.264172 gal | |
| Time | second (s) | minute (min) | 60 s = 1 min |
Tip: Always write the unit after the number. Keep track of powers when converting areas (m² → ft²) or volumes (m³ → ft³).
3. Length, Perimeter & Circumference
3.1 Straight line length
- Direct measurement with a ruler/metric tape.
- In coordinate geometry:
[ \text{Length of } AB = \sqrt{(x_B-x_A)^2+(y_B-y_A)^2} ]
3.2 Perimeter of polygons
- Sum of side lengths.
- For a regular n‑gon of side (s):
[ P = ns ]
3.3 Circumference of a circle
- Derivation:
[ C = 2\pi r = \pi d ] (comes from the definition (C = \pi d) or from the limit of perimeters of inscribed polygons).
4. Area of Basic Shapes
| Shape | Formula | Derivation / Key Points |
|---|---|---|
| Rectangle | (A = lw) | Simple product of adjacent sides. |
| Square | (A = s^2) | Special case of rectangle. |
| Triangle | (A = \frac{1}{2} bh) | Base × height / 2. Height is perpendicular to base. |
| Isosceles triangle | (A = \frac{1}{4} a \sqrt{4b^2-a^2}) | Using Pythagoras to find height. |
| Trapezium | (A = \frac{1}{2}(a+b)h) | Average of parallel sides times height. |
| Parallelogram | (A = bh) | Base × height (height is perpendicular to base). |
| Rhombus | (A = \frac{1}{2} d_1 d_2) | Half the product of diagonals. |
| Circle | (A = \pi r^2) | Derived by integration or by (C = \pi d) and (A = \frac{1}{2} r C). |
| Sector of a circle | (A = \frac{\theta}{360^\circ}\pi r^2) | Proportional to central angle. |
4.1 Tricky Points
- Height in a triangle: Always