GRADE 9 Mathematics MEASUREMENTS – MASS,VOLUME,WEIGHT AND DENSITY Notes
MEASUREMENTS — MASS, VOLUME, WEIGHT & DENSITY
Subject: Mathematics | Subtopic: Mass, Volume, Weight and Density | Age: 14 (Kenyan syllabus)
1. Quick overview
These four ideas help us describe how much matter an object has (mass), how much space it takes (volume), the pull of gravity on it (weight) and how compact the matter is (density).
2. Definitions and units
- Mass (m): amount of matter in an object. Measured in kilograms (kg) or grams (g). Use a balance (beam balance) to compare masses.
- Volume (V): space taken by an object. Units: cubic metre (m³), cubic centimetre (cm³), litre (L). For regular solids we use formulas; for liquids use a measuring cylinder.
- Weight (W): force due to gravity on a mass. Measured in newtons (N). Related to mass by W = m × g where g (gravity) ≈ 9.8 N/kg (approx 10 N/kg often used in classwork).
- Density (ρ): mass per unit volume. ρ = m ÷ V. Common units: kg/m³ or g/cm³. Water has ρ ≈ 1 g/cm³ = 1000 kg/m³.
3. Useful unit conversions
- 1 kg = 1000 g
- 1 m³ = 1000 L
- 1 L = 1 dm³ = 1000 cm³
- 1 g/cm³ = 1000 kg/m³
4. Important formulas
Mass: m = given (kg or g) | Volume: use shape formulas (below) | Weight: W = m × g | Density: ρ = m ÷ V
- Cube: V = a³ (a = side length)
- Rectangular block: V = length × width × height
- Cylinder: V = πr²h
- Sphere: V = 4/3 πr³ (you may not need this until higher topics)
5. Visuals — simple diagrams
6. Worked examples (step-by-step)
A stone has mass 300 g and volume 100 cm³. Find its density in g/cm³ and kg/m³.
Step 1: ρ = m ÷ V = 300 g ÷ 100 cm³ = 3 g/cm³.
Step 2: Convert to kg/m³: 1 g/cm³ = 1000 kg/m³, so 3 g/cm³ = 3000 kg/m³.
A bag of maize has mass 25 kg. Find its weight on Earth (use g = 10 N/kg for easy calculation).
W = m × g = 25 kg × 10 N/kg = 250 N.
(If precise g = 9.8 N/kg, W = 245 N.)
A wooden block measures 0.4 m × 0.25 m × 0.1 m. Find its volume in m³ and litres.
V = 0.4 × 0.25 × 0.1 = 0.01 m³.
Convert to litres: 1 m³ = 1000 L, so 0.01 m³ = 10 L.
7. Typical densities (useful to remember)
- Water: 1.00 g/cm³ = 1000 kg/m³
- Milk: ≈1.03 g/cm³
- Iron: ≈7.87 g/cm³
- Air (at sea level): ≈1.2 kg/m³ (very small compared to solids)
8. Practice questions
- A metal cube has side 2 cm and mass 62.96 g. Find its density in g/cm³.
- A liquid of density 0.8 g/cm³ is poured into a 5 L container. What mass of the liquid (in kg) does the container hold when full?
- A cylinder has radius 5 cm and height 10 cm. Find its volume in cm³. If its mass is 6.28 kg, find its density in g/cm³. (Use π ≈ 3.14)
- A student measures mass = 120 g on a balance and then measures weight on a spring scale. What weight (N) should the student expect on Earth (use g = 9.8 N/kg)?
9. Tips for exam success
- Always check and convert units before substituting (kg vs g, cm³ vs m³).
- For quick classroom calculations use g = 10 N/kg if the question allows approximation.
- Remember 1 cm³ = 1 mL; so volumes in cm³ can often be read as millilitres for liquids.
- If density > 1 g/cm³, object sinks in water; if < 1 g/cm³, it floats.