Arcs, sectors and mensuration
Arc lengths, sector areas, and the surface areas and volumes of prisms, cylinders, pyramids, cones and spheres, including frustums and similar solids. A solid-figure question is common in DSE Paper 1.
What you will learn
- 16.1–16.2 Arc length and sector area formulas; arcs are proportional to the angle at the centre.
- 16.3 Perimeters and areas of composite figures.
- 18.1 Volumes of prisms, cylinders, pyramids, cones and spheres; height of a 3-D figure.
- 18.2 Surface areas of right prisms, cylinders, right pyramids, right cones and spheres.
- 18.3 Similar figures: lengths, areas and volumes; frustums.
- 18.4 Problems with volume and surface area.
You need first: Geometry and data basics, Similar triangles
Key terms
| English | 中文 |
|---|---|
| arc, arc length | 弧，弧長 |
| sector | 扇形 |
| angle at the centre | 圓心角 |
| prism, pyramid | 角柱，角錐 |
| cylinder, cone, sphere, hemisphere | 圓柱，圓錐，球體，半球體 |
| frustum | 平截頭體 |
| slant height | 斜高 |
| curved surface area, total surface area | 曲面面積，總表面面積 |
| volume, capacity | 體積，容量 |
| similar solids | 相似立體 |
Standard steps
Sectors: a fraction of a circle
Fraction = θ / 360°.
- Arc length = 2πr × θ/360°
- Sector area = πr² × θ/360°
- Sector perimeter = arc + 2r
Formula sheet
| Solid | Volume | Surface |
|---|---|---|
| Prism, cylinder | base area × height | cylinder curved surface = 2πrh |
| Pyramid, cone | ⅓ × base area × height | cone curved surface = πrl (l = slant height) |
| Sphere | (4/3)πr³ | 4πr² |
Why the ⅓: three equal pyramids fill a cube. A cone is a "round pyramid".
Similar figures
If the scale factor of lengths is k:
- area ratio = k²;
- volume ratio = k³.
Frustum
Volume of frustum = volume of the big cone − volume of the small cone cut off the top. The small cone is similar to the big one.
- Complete the cone. Find the height (or radius) of the small cone with similar triangles.
- Work out both cone volumes, V = ⅓πr²h.
- Subtract: frustum = big cone − small cone. Check with the volume ratio = (scale factor)³.
Melting and recasting
The volume stays the same. Set the two volumes equal and solve.
- Write the volume of the original solid.
- Write the total volume of the new solid or solids, with the unknown.
- Set them equal and solve. Volume does not change when a solid is melted, but surface area does.
Worked examples
arc = 2π(9) × 80/360 = 4π ≈ 12.6 cm
area = π(9²) × 80/360 = 18π ≈ 56.5 cm²
perimeter = 4π + 18 ≈ 30.6 cmslant height l = √(6² + 8²) = 10 cm
volume = ⅓π(6²)(8) = 96π cm³
curved surface = π(6)(10) = 60π cm²small cone : big cone, length ratio = 1 : 2
volume ratio = 1 : 8
frustum = big − small = 96π × (1 − 1/8) = 84π cm³volume ratio = (4/6)³ = 8 : 27
larger volume = 40 × 27/8 = 135 cm³Common mistakes
- Using the vertical height instead of the slant height in πrl, or the reverse in the volume.
- Forgetting the ⅓ for cones and pyramids.
- Using the length ratio for volumes (it must be cubed).
- In the frustum, using the cut-off height for the small cone when the question gives the frustum height.
- Missing the bottom (or top) when finding total surface area.
- Mixing units: cm³ and m³, or capacity in L (1 L = 1000 cm³).
How the DSE tests it
- A(2): a multi-part solids question, often with a frustum, similar solids or recasting, and a final "explain" part that compares two values.
- Paper 2: sector areas, composite figures, and volume ratios of similar solids.
Questions you might not ask in class
Slant height or height: how do I know?
Height goes straight up, at right angles to the base. The slant height runs along the sloping surface. Volume uses height; the curved surface uses slant height.
Why is the frustum's volume not just 'half the cone'?
The bottom part is fatter. Cutting halfway up removes only 1/8 of the volume.
What does 'melted and recast' mean?
The metal is melted down and reshaped. Nothing is lost, so the volume is the same.