Areas Related to Circles, Surface Areas and Volumes
Areas Related to Circles and Surface Areas and Volumes supplies formula-driven 1-mark and 3-mark questions, plus a combination-of-solids problem worth 4–5 marks.
Circle measures
Circumference 2πr, area πr². For a sector of angle θ: arc length = (θ/360)·2πr and area = (θ/360)·πr². Area of the minor segment = sector area − area of the triangle formed by the two radii and the chord. Area of a ring = π(R² − r²). Use π = 22/7 when radii are multiples of 7, otherwise 3.14.
Solids
- Cylinder: CSA 2πrh, TSA 2πr(r + h), volume πr²h.
- Cone: l = √(r² + h²), CSA πrl, TSA πr(l + r), volume ⅓πr²h.
- Sphere: surface 4πr², volume (4/3)πr³; hemisphere TSA 3πr², volume (2/3)πr³.
- Frustum: CSA π(r₁ + r₂)l with l = √[h² + (r₁ − r₂)²], volume ⅓πh(r₁² + r₁r₂ + r₂²).
For a combination, add only exposed surfaces and add the volumes of the parts.
Worked example
A cone has radius 7 cm and height 24 cm. Find its volume (and slant height). l = √(7² + 24²) = √(49 + 576) = 25 cm. V = ⅓ × (22/7) × 7² × 24 = ⅓ × 22 × 7 × 24 = 1232 cm³.
Traps that cost marks
- Diameter given instead of radius - halve it before substituting.
- Mixing cm and m: convert first, then give the answer in the requested unit (1000 cm³ = 1 litre).
- In combinations the hidden circular face must not be counted; a hemisphere on a cylinder contributes 2πr² of curved surface, not 3πr².
- A segment needs the triangle subtracted from the sector, never added.