GRADE 9 Mathematics MEASUREMENTS – VOLUME OF SOLIDS Notes
Mathematics — Measurements
Subtopic: Volume of Solids (Target age: 14, Kenya)
Learning goals:
- Know the formulas for volumes of common solids (cube, cuboid, cylinder, cone, sphere, prism).
- Use units correctly and convert between cm³, m³ and litres.
- Solve routine problems and simple composite-solid problems.
Units and conversions
- Basic volume units: cubic centimetre (cm³), cubic metre (m³), litre (L), millilitre (mL).
- 1 cm³ = 1 mL. 1 m³ = 1000 L. 1 m³ = 1 000 000 cm³.
- Always convert all measurements to the same unit before calculating volume.
Common solids, formulas and quick visuals
Area of triangle = 1/2 × base × height
Worked examples
A water tank is a cylinder with radius 0.5 m and height 2.5 m. Find its volume in litres. (Take π = 3.1416)
V = π r² h = 3.1416 × (0.5)² × 2.5 = 3.1416 × 0.25 × 2.5 = 1.9635 m³ ≈ 1.9635 m³.
Convert to litres: 1 m³ = 1000 L → 1.9635 × 1000 = 1963.5 L.
A school dustbin is shaped like a cylinder of radius 0.3 m and height 0.6 m with a hemisphere lid of the same radius. Find total volume (in m³). Use π = 22/7.
Cylinder Vc = π r² h = (22/7) × 0.09 × 0.6 = (22/7) × 0.054 = 0.169714... m³ ≈ 0.1697 m³.
Hemisphere Vh = (1/2) × (4/3)πr³ = (2/3)π r³ = (2/3)(22/7) × 0.027 = (44/21) × 0.027 ≈ 0.05657 m³.
Total ≈ 0.1697 + 0.0566 = 0.2263 m³ ≈ 226.3 L.
Exam tips & common mistakes
- Always check and convert units first. If lengths are in cm, volume will be in cm³ unless converted.
- Use radius (r) in formulas with r² or r³ — if given diameter (d), r = d/2.
- Decide whether to use π = 22/7 or 3.1416 depending on the question instructions; show working.
- For composite solids, split into simple solids, find each volume and add (or subtract) as required.
Practice questions (try before checking answers)
- Find the volume of a cuboid 25 cm by 10 cm by 8 cm. Give answer in cm³.
- A cone has radius 5 cm and height 12 cm. Find its volume. Use π = 3.1416.
- A spherical water ball has diameter 14 cm. Find the volume. Give answer in litres (1 L = 1000 cm³). Use π = 3.1416.
- A swimming pool is 10 m long, 4 m wide and 1.8 m deep. How many litres of water will fill it?
- Mix: A cylindrical well radius 0.8 m depth 3 m has a hemispherical bottom of radius 0.8 m. Find total volume in m³ (use π = 22/7).
Answers (expand to view)
- V = 25×10×8 = 2000 cm³.
- V = 1/3 π r² h = (1/3)×3.1416×25×12 = (1/3)×3.1416×300 = 314.16 cm³.
- r = 7 cm, V = (4/3)π r³ = (4/3)×3.1416×343 ≈ 1436.76 cm³ ≈ 1.437 L.
- V = 10×4×1.8 = 72 m³ → 72×1000 = 72 000 L.
- Cylinder: Vc = π r² h = (22/7)×0.64×3 = (22/7)×1.92 ≈ 6.0342857 m³. Hemisphere: Vh = 2/3 π r³ = (2/3)(22/7)×0.512 ≈ 0.3215238 m³. Total ≈ 6.356 m³.
Quick checklist before answering volume questions
- Are the measurements in the same unit? If not, convert.
- Is the formula correct for the solid? (Use r for radius.)
- State the value of π being used.
- Write units with your final answer (cm³, m³, L).
Good practice: sketch the solid, label dimensions, split composite shapes into simple ones, and compute step by step.