Mensuration
Mensuration
Mensuration tests perimeter, area, surface area and volume of plane and solid shapes, including circles, sectors, prisms, cylinders, cones, spheres and composite figures.
Core rules
- Circle: circumference = 2πr, area = πr². Sector arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².
- Triangle area = ½ × base × perpendicular height; parallelogram = base × height; trapezium = ½(a + b)h.
- Volume of a prism = cross-sectional area × length. Cylinder volume = πr²h; cone = ⅓πr²h; sphere = 4/3πr³.
- Curved surface area of a cylinder = 2πrh; of a cone = πrl, where l is the slant height; sphere surface area = 4πr².
- For similar solids, lengths scale by k, areas by k² and volumes by k³.
Worked example
A sector has radius 9 cm and angle 80°. Its area is (80/360) × π × 9² = (2/9) × 81π = 18π ≈ 56.5 cm². Its arc length is (80/360) × 2π × 9 = 4π ≈ 12.6 cm. Keep π in the working until the final decimal to avoid rounding error.
Traps
- Using diameter instead of radius in circle formulas.
- Confusing arc length with sector area, or slant height with perpendicular height.
- Adding areas of all faces when a surface-area question asks for the curved surface only.
- Mixing units, such as cm and m, before calculating.