Quick revision notes · Mathematics 4024 · Chapter 9 of 35
Mensuration
Units, areas and perimeters, circles and sectors, surface area and volume of every solid, and working backwards.
At a glance
Units, areas and perimeters, circles and sectors, surface area and volume of every solid, and working backwards.
Volumes of a cone, pyramid and sphere, and the curved surface areas of a cone and sphere, are on the formula list; the rest are not.
1Units
- Length: 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm.
- Area: square the conversion — 1 m² = 1002 = 10 000 cm²; 1 cm² = 100 mm².
- Volume: cube it — 1 m³ = 1003 = 1 000 000 cm³.
- Capacity: 1 litre = 1000 cm³ = 1000 ml; 1 cm³ = 1 ml.
| Convert | Length | Area | Volume |
|---|---|---|---|
| m → cm | × 100 | × 10 000 | × 1 000 000 |
| cm → mm | × 10 | × 100 | × 1000 |
| km → m | × 1000 | × 1 000 000 | — |
Worked example
Convert 3.5 m² to cm².
- 1 m² = 10 000 cm².
- 3.5 × 10 000.
35 000 cm²
2Area and perimeter of 2D shapes
- Perimeter = distance all the way round. Area = space inside.
- Use the perpendicular height for triangles, parallelograms and trapezia.
- Compound shapes: split into rectangles, triangles and semicircles, then add (or subtract a hole).
Formulas
Worked example
A semicircle of diameter 8 cm sits on top of an 8 cm × 5 cm rectangle. Find the total area.
- Rectangle: 8 × 5 = 40 cm².
- Semicircle: ½ × π × 42 = 8π = 25.13 cm².
65.1 cm² (3 s.f.)
3Circles, arcs, sectors and segments
- Circumference = πd = 2πr. Area = πr2.
- A sector is θ/360 of the circle: arc length = θ/360 × 2πr; sector area = θ/360 × πr2.
- Perimeter of a sector = arc + 2 radii.
- Segment area = sector area − triangle area (triangle = ½r2 sin θ).
Formulas
Worked example
A sector has radius 10 cm and angle 90°. Find the area of the minor segment.
- Sector: ¼ × π × 100 = 78.54 cm².
- Triangle: ½ × 10 × 10 = 50 cm².
- 78.54 − 50.
28.5 cm² (3 s.f.)
Exam tip: 'In terms of π' means leave π in: the sector area 12π, not 37.7.
4Prisms, cuboids and cylinders
- Volume of a prism = area of cross-section × length. A cylinder is a prism with a circular cross-section.
- Surface area = total area of all faces (draw the net if unsure).
- Cylinder curved surface = 2πrh (the rectangle that wraps round); total = 2πrh + 2πr2.
Formulas
Worked example
A cylinder has radius 3 cm and height 10 cm. Find its volume and total surface area in terms of π.
- V = π × 9 × 10 = 90π cm³.
- Curved: 2π × 3 × 10 = 60π; two ends: 2 × 9π = 18π.
V = 90π cm³; SA = 78π cm²
5Pyramids, cones and spheres
- Pyramid and cone volume = ⅓ × base area × height.
- Cone curved surface = πrl (l = slant height; l2 = r2 + h2).
- Sphere: V = ⁴⁄₃πr3, SA = 4πr2. Hemisphere: half of each, plus the flat circle πr2 for total SA.
- Frustum = big cone − small cone.
Formulas
Worked example
A cylinder (radius 2 cm, height 5 cm) has a hemisphere of radius 2 cm on top. Find the volume.
- Cylinder: π × 4 × 5 = 20π.
- Hemisphere: ½ × ⁴⁄₃ × π × 8 = 16π/3.
- Total = 76π/3 = 79.6.
79.6 cm³ (3 s.f.)
Exam tip: The formulas for cone, pyramid and sphere are given on the formula list — but you must know which to use.
6Working backwards
- Given the volume or area, substitute and solve for the missing length.
- Rearrange before using the calculator: r2 = V ÷ (πh), then square root.
- Equal volumes: liquid poured from one container into another keeps its volume.
Worked example
A cylinder has volume 500 cm³ and height 8 cm. Find its radius.
- 500 = πr2 × 8.
- r2 = 500 ÷ (8π) = 19.89…
- r = √19.89…
4.46 cm (3 s.f.)
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Using the diameter as the radius | Halve the diameter first |
| Cone volume πr2 h | It's ⅓πr2 h |
| 1 m² = 100 cm² | 1 m² = 10 000 cm² |
| Sector perimeter = arc length only | Add the two radii |
| Using the slant height as h in a cone's volume | Volume uses the vertical height |
★In the exam
- Write the formula, substitute, then calculate — each step can earn a mark.
- Keep π in until the end (or use the π key), then round.
- Check units: area in cm², volume in cm³.