Areas Related to Circles and Surface Areas and Volumes
This domain tests sector and segment areas, arc length, and the surface areas and volumes of cylinders, cones, spheres, hemispheres and frustums, including combinations of solids.
Core rules
- Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr².
- Segment area = sector area − triangle area.
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h), volume = πr²h.
- Cone: slant height l = √(r² + h²), CSA = πrl, TSA = πr(r + l), volume = ⅓πr²h.
- Sphere: surface area = 4πr², volume = 4/3πr³; hemisphere: TSA = 3πr².
- Frustum: volume = ⅓πh(r₁² + r₁r₂ + r₂²), CSA = π(r₁ + r₂)l.
- For combined solids, add exposed surfaces only and add volumes of joined parts.
Worked example
A cone has radius 3 cm and height 4 cm. Find its slant height, curved surface area and volume. l = √(3² + 4²) = 5 cm. CSA = πrl = π × 3 × 5 = 15π cm². Volume = ⅓πr²h = ⅓π × 9 × 4 = 12π cm³. If the cone is surmounted on a cylinder, the base circle of the cone is hidden, so subtract it from the total exposed area.
Traps
- Using height instead of slant height in a cone's CSA.
- Mixing centimetres and metres.
- Counting hidden circular faces when solids are joined.
- Converting 1000 cm³ to 1 litre incorrectly.