Mensuration (Areas, Surface Areas, Volumes)
This domain tests areas of sectors and segments, surface areas and volumes of cylinders, cones, spheres, hemispheres and frustums, and solids formed by combining these shapes.
Core rules
- Cylinder: curved surface 2πrh, total 2πr(r + h), volume πr²h.
- Cone: slant height l = √(r² + h²), curved surface πrl, total πr(l + r), volume ⅓πr²h.
- Sphere: surface 4πr², volume 4/3πr³; hemisphere: curved 2πr², total 3πr², volume 2/3πr³.
- Frustum: volume ⅓πh(r₁² + r₁r₂ + r₂²), slant height √[h² + (r₁ − r₂)²].
- Sector area = (θ/360)πr²; arc length = (θ/360)2πr; segment = sector − triangle.
- For combinations, add only exposed surfaces and add volumes of joined parts.
Worked example
A cylinder has radius 7 cm and height 10 cm. Volume = πr²h = (22/7) × 49 × 10 = 1540 cm³. Total surface area = 2πr(r + h) = 2 × (22/7) × 7 × 17 = 748 cm².
Traps
Using diameter in place of radius; adding the hidden circular face where two solids join; confusing curved and total surface area; mixing cm and m without converting; using π = 22/7 when the radius is not a multiple of 7.