Mensuration (Areas, Surface Areas and Volumes)
Mensuration is the most formula-dependent domain: cylinder, cone, sphere, hemisphere and frustum, often combined into a single solid such as a capsule or a tent. Accuracy in identifying radius, height and slant height decides the answer.
Core rules
- Cylinder: curved surface 2πrh, total surface 2πr(h + r), volume πr²h.
- Cone: slant height l = √(r² + h²), curved surface πrl, total surface πr(l + r), volume ⅓πr²h.
- Sphere: surface 4πr², volume (4/3)πr³. Hemisphere: curved surface 2πr², total surface 3πr², volume (2/3)πr³.
- Frustum: volume ⅓πh(r₁² + r₂² + r₁r₂), slant height √[h² + (r₁ − r₂)²].
- Combined solids: volumes add; for surfaces add only the exposed portions. 1000 cm³ = 1 litre.
Worked example
A cone has radius 7 cm and height 24 cm. Find its curved surface area and volume (π = 22/7). l = √(7² + 24²) = √625 = 25 cm. Curved surface = πrl = (22/7)(7)(25) = 550 cm². Volume = ⅓πr²h = ⅓(22/7)(49)(24) = 22 × 7 × 8 = 1232 cm³.
Traps
- Substituting the diameter for the radius.
- Using the vertical height where the slant height is required in πrl.
- Counting the hidden circular face when a hemisphere sits on a cylinder.
- Mixing units, for example leaving a tank capacity in cm³ when litres are asked.
- Rounding π early and then rounding again at the end.