Mensuration and Geometry
Mensuration and geometry covers the measurement of plane figures and solids plus the basic angle and triangle facts that support them. The examiner typically gives dimensions of a plot, tank, cone or pipe and asks for area, perimeter, volume, capacity or surface area - often with a unit conversion built in.
Core rules
- Rectangle: area lb, perimeter 2(l + b). Triangle: area ½ × base × height, or √[s(s − a)(s − b)(s − c)] with s the semi-perimeter.
- Circle: area πr², circumference 2πr. Trapezium: ½(a + b)h.
- Cylinder: volume πr²h, curved surface 2πrh, total 2πr(r + h).
- Cone: slant height l = √(r² + h²), volume ⅓πr²h, curved surface πrl, total πr(l + r).
- Sphere: volume (4/3)πr³, surface 4πr²; hemisphere: (2/3)πr³ and 3πr².
- Angle sum of a polygon = (n − 2) × 180°; Pythagoras holds for right triangles, with triples such as 3-4-5 and 5-12-13.
Worked example
A conical heap has radius 3 m and height 4 m. First find the slant height: l = √(3² + 4²) = 5 m. Volume = ⅓π(3²)(4) = 12π ≈ 37.7 m³. Curved surface = πrl = π × 3 × 5 = 15π ≈ 47.1 m². Total surface = πr(l + r) = π × 3 × 8 = 24π ≈ 75.4 m².
Traps
Substituting the diameter for the radius; using height where slant height belongs; converting 1 m³ incorrectly (1 m³ = 1,000 litres = 10⁶ cm³); and quoting curved surface when total surface area is asked.