Mensuration
Mensuration
Mensuration tests the measurement of length, area, surface area and volume - including arcs, sectors and composite solids - and the conversion of units. Every answer needs its unit, and compound shapes usually require you to add or subtract two standard results.
Plane figures
- Circle: circumference = πd = 2πr; area = πr².
- Trapezium: area = ½(a + b)h. Parallelogram: bh. Triangle: ½bh.
Arcs and sectors
- Arc length = θ/360 × 2πr.
- Sector area = θ/360 × πr².
Solids
- Cuboid: V = lbh; surface area = 2(lb + lh + bh).
- Cylinder: V = πr²h; curved surface = 2πrh.
- Cone: V = ⅓πr²h; curved surface = πrl, where l is the slant height and l² = r² + h².
- Sphere: V = 4/3 πr³; surface area = 4πr².
- Prism: V = area of cross-section × length. Pyramid: V = ⅓ × base area × height.
Units
- 1 m = 100 cm, so 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³. Also 1 litre = 1000 cm³.
Worked example. A sector has radius 12 cm and angle 150°. Find its arc length and area. Arc = 150/360 × 2 × π × 12 = 5/12 × 24π = 10π ≈ 31.4 cm. Area = 150/360 × π × 12² = 5/12 × 144π = 60π ≈ 188.5 cm².
Traps that cost marks.
- Substituting the diameter where the formula needs the radius.
- Using the perpendicular height instead of the slant height for a cone's curved surface.
- Forgetting the two circular ends when a cylinder is described as closed.
- Converting cm² to m² by dividing by 100 instead of 10 000.