Measurement and Geometry - Te Ine me te Āhuahanga
Measurement and geometry questions test whether you can choose units, calculate lengths, areas, volumes and angles, and use scale and Pythagoras in practical shapes.
Core rules and concepts
- Metric length conversions step by 10: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m. Area units step by 100 (1 cm² = 100 mm²); volume units step by 1000 (1 m³ = 1 000 000 cm³).
- Capacity: 1 L = 1000 mL = 1000 cm³; 1 m³ = 1000 L.
- Perimeter is the distance around a shape. Circle circumference C = πd = 2πr.
- Area formulas: rectangle l × w; triangle ½bh; circle πr². Volume of a prism = area of cross-section × length.
- Angle facts: angles on a straight line sum to 180°, at a point 360°, in a triangle 180°, in a quadrilateral 360°. Parallel lines give equal corresponding and alternate angles.
- Pythagoras: for a right triangle, a² + b² = c², where c is the hypotenuse.
- Scale 1:100 means 1 cm on the drawing represents 100 cm in real life. Lengths scale by the factor, areas by its square, volumes by its cube.
Worked example
A cylindrical water tank has radius 0.8 m and height 1.5 m. Find its volume in litres. V = πr²h = π × 0.8² × 1.5 = π × 0.64 × 1.5 ≈ 3.016 m³. Since 1 m³ = 1000 L, volume ≈ 3016 L. Round to a sensible 3020 L if the context allows. The radius was given, so do not halve it again.
Traps
- Using the diameter as the radius.
- Mixing mm, cm and m in one calculation without converting first.
- Forgetting that area scale factors are squared and volume factors cubed.
- Using the sloping side as the hypotenuse when it is not opposite the right angle.