Measurement and Geometry: units, area, volume, Pythagoras and scale
Measurement and geometry questions in the numeracy CAT are nearly always practical: how much paint, how many tiles, will the ladder fit, what does this plan mean. The assessment tests unit conversion, perimeter and area of composite shapes, volume of prisms, Pythagoras, angle reasoning and scale factors.
Core rules
- Metric prefixes: kilo = ×1000, centi = ÷100, milli = ÷1000; 1 cm³ = 1 mL and 1 m³ = 1000 L.
- Area units are squared, so 1 m² = 10 000 cm², not 100.
- Rectangle A = lw; triangle A = ½bh; circle A = πr² and C = 2πr, where r is the radius (half the diameter).
- Prism volume V = area of cross-section × length; cylinder V = πr²h.
- Pythagoras: a² + b² = c², where c is the longest side (the hypotenuse).
- Angles: triangle 180°, straight line 180°, point 360°. Similar shapes scale lengths by k, areas by k² and volumes by k³.
Worked example
A deck measures 4.2 m by 3.5 m and needs two coats of stain. One litre covers 8 m². How many 1 L tins are needed, at $23.50 each?
Deck area = 4.2 × 3.5 = 14.7 m². Two coats = 2 × 14.7 = 29.4 m². Litres = 29.4 ÷ 8 = 3.675 L. Tins come only in whole litres, so round up to 4 tins, costing 4 × 23.50 = $94.00. Rounding down to 3 tins would leave the second coat 0.325 L short.
Traps that cost marks
- Rounding down on ‘how many containers, people or trips’ questions - always round up.
- Substituting the diameter where the formula needs the radius.
- Converting cm² to m² by dividing by 100 instead of 10 000.
- Forgetting to subtract a doorway, window or gap when finding the area to be covered.
- Applying a linear scale factor directly to an area or volume.