Mensuration and Trigonometry: solids, ratios and identities
Mensuration and Trigonometry (WBBSE Madhyamik)
This domain pairs solid mensuration - cuboid, right circular cylinder, cone, sphere - with trigonometric ratios, complementary angles and identities. Mensuration questions are volume and surface problems, often phrased as melting and recasting; trigonometry questions are evaluations, simplifications and identity proofs done without tables.
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 (1/3)πr²h.
- Sphere: surface 4πr², volume (4/3)πr³; hemisphere total surface 3πr².
- Recasting conserves volume: number of small objects = volume of the large object ÷ volume of one small object.
- Radian measure: π radians = 180°; arc length = rθ with θ in radians.
- Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.
- Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
Worked example
A cone has r = 7 cm and h = 24 cm. Then l = √(49 + 576) = √625 = 25 cm. Volume = (1/3)(22/7)(49)(24) = 1232 cm³, and curved surface = (22/7)(7)(25) = 550 cm².
Traps
- Substituting the diameter where the formula needs the radius.
- Omitting the factor 1/3 in cone volume, or giving curved surface when total surface is asked.
- Mixing degrees and radians inside one expression.
- 'Proving' an identity by altering both sides at once instead of working from one side to the other.