Mensuration: Surface Areas, Volumes and Areas Related to Circles
Mensuration items combine solids (cylinder, cone, sphere, hemisphere, frustum) with circle parts (sector, segment) and often involve conversion of units or a solid recast into another shape.
Solids
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h.
- Cone: l = √(r² + h²), CSA = πrl, TSA = πr(l + r), V = (1/3)πr²h.
- Sphere: surface area 4πr², volume (4/3)πr³. Hemisphere: curved area 2πr², TSA 3πr², volume (2/3)πr³.
- Frustum: V = (1/3)πh(r₁² + r₂² + r₁r₂), CSA = πl(r₁ + r₂) with l = √(h² + (r₁ − r₂)²).
Areas related to circles
- Sector area = (θ/360) × πr²; arc length = (θ/360) × 2πr.
- Segment area = sector area − triangle area (for θ < 180°).
Worked example
A cone has radius 7 cm and height 24 cm. Find its curved surface area (π = 22/7). l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm. CSA = πrl = (22/7) × 7 × 25 = 550 cm². The base is excluded because only the curved surface is asked.
Traps that cost marks
- Substituting the diameter for the radius.
- Giving TSA when CSA is asked; a cone's TSA includes the base circle.
- Forgetting 1 litre = 1000 cm³ when a capacity is requested.
- Using height instead of slant height in a cone's curved area, or adding radii instead of using slant height in a frustum.