Units, Area and Perimeter at Extended Level
Square and cube the conversion — then work backwards
Quick recap — length. 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km. Mass: 1000 g = 1 kg, 1000 kg = 1 tonne. Capacity: 1000 ml = 1 litre.
1 m² = 100 × 100 = 10 000 cm² 1 m³ = 100 × 100 × 100 = 1 000 000 cm³
Also 1 cm² = 100 mm², and 1 cm³ = 1000 mm³.
Capacity link: 1 cm³ = 1 ml, 1000 cm³ = 1 litre, 1 m³ = 1000 litres.
Compound units join two units, such as km/h, m/s or g/cm³. Convert each part on its own. For example \(72\) km/h \(= \dfrac{72 \times 1000}{3600}\) m/s \(= 20\) m/s.
Area formulas (height is always the perpendicular height, at right angles to the base):
- Rectangle: \(A = lw\) Parallelogram: \(A = bh\) Triangle: \(A = \dfrac{1}{2}bh\)
- Trapezium: \(A = \dfrac{1}{2}(a + b)h\), where \(a\) and \(b\) are the parallel sides.
What Extended adds: reverse problems (you know the area, find a length), sides written in algebra (form an equation and solve it — sometimes a quadratic), and mixing unit changes into area and volume questions. Perimeter is the total distance round the outside — add every side, including ones you have to work out.
Convert (a) 3.6 m² to cm², (b) 250 000 cm³ to m³ and to litres.
(a) \(3.6 \times 10\,000 = 36\,000\) cm²
(b) \(250\,000 \div 1\,000\,000 = 0.25\) m³ and \(250\,000 \div 1000 = 250\) litres
Bigger unit → smaller number, so divide. Smaller unit → bigger number, so multiply.
A water tank in a Maun backyard is a cuboid 1.2 m long, 80 cm wide and 50 cm high. How many litres does it hold when full?
Change everything to cm first: 1.2 m = 120 cm.
\(V = 120 \times 80 \times 50 = 480\,000\) cm³ \(= \dfrac{480\,000}{1000} = 480\) litres
Mixing metres and centimetres in one multiplication is the classic error. Pick one unit before you start.
(a) Change 72 km/h to m/s. (b) Wood has density 0.8 g/cm³. Write this in kg/m³.
(a) \(72\) km/h \(= \dfrac{72\,000 \text{ m}}{3600 \text{ s}} = 20\) m/s
(b) 1 m³ = 1 000 000 cm³, so 1 m³ of wood has mass \(0.8 \times 1\,000\,000 = 800\,000\) g \(= 800\) kg. Density \(= 800\) kg/m³.
The trapezium below has parallel sides \(x\) cm and \((x + 6)\) cm and height 5 cm. Its area is 60 cm². Find \(x\).
\(\dfrac{1}{2}(x + x + 6)(5) = 60 \Rightarrow \dfrac{5(2x + 6)}{2} = 60 \Rightarrow 5x + 15 = 60 \Rightarrow x = 9\)
Check: parallel sides 9 and 15, \(\dfrac{1}{2}(9 + 15) \times 5 = 60\) ✔.
A rectangle has length \((2x + 3)\) cm and width \((x – 1)\) cm. Its perimeter is 40 cm. Find its area.
\(2(2x + 3 + x – 1) = 40 \Rightarrow 6x + 4 = 40 \Rightarrow x = 6\)
Length \(= 15\) cm, width \(= 5\) cm, so area \(= 15 \times 5 = 75\) cm². The question asked for the area — finding \(x\) is only halfway.
Q1. Change 4.5 m² into cm².
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Q2. A container holds 0.07 m³. Write this in (a) cm³, (b) litres.
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(b) 70 000 ÷ 1000 = 70 litres ✓ B1
Q3. A wall is 3.2 m long and 2.5 m high. It is covered with square tiles of side 20 cm. How many tiles are needed?
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8 ÷ 0.04 = 200 tiles ✓ A1
Q4. A cuboid water tank measures 1.5 m by 1.2 m by 90 cm. It is filled at 12 litres per minute. How long does it take to fill the empty tank? Give your answer in hours and minutes.
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1620 ÷ 12 = 135 minutes ✓ M1
= 2 hours 15 minutes ✓ A1
Q5. A combi travels between Gaborone and Mochudi at an average speed of 54 km/h. Write this speed in m/s.
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= 15 m/s ✓ A1
Q6. A parallelogram has base \((x + 5)\) cm and perpendicular height \(x\) cm. Its area is 36 cm². Find \(x\).
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(x + 9)(x − 4) = 0 ✓ M1
x = 4 (x = −9 is impossible for a length) ✓ A1
Q7. The trapezium below has parallel sides 8 cm and 14 cm. Its area is 77 cm². Find its height \(h\).
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h = 7 cm ✓ A1
Q8. A triangle has sides \((x + 2)\) cm, \((2x – 1)\) cm and \((x + 5)\) cm. Its perimeter is 30 cm. Find the length of its shortest side.
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x = 6 ✓ A1
sides 8, 11, 11 → shortest = 8 cm ✓ A1
Q9. A block of wood measures 30 cm by 20 cm by 5 cm and has a mass of 2.1 kg. Find its density in (a) g/cm³, (b) kg/m³.
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(a) 2100 ÷ 3000 = 0.7 g/cm3 ✓ A1
(b) 0.7 × 1 000 000 ÷ 1000 = 700 kg/m3 ✓ B1
Q10. A field at a cattle post is a trapezium. Its parallel sides are 120 m and 80 m, and the perpendicular distance between them is 50 m. Grass seed costs P35 for every 100 m². Find the cost of seeding the whole field.
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5000 ÷ 100 × 35 = P1750 ✓ A1
- Using the length factor for area or volume: m² → cm² is × 10 000, not × 100; m³ → cm³ is × 1 000 000.
- Mixing units in one calculation: change every length to the same unit before you multiply.
- Using the slant side as the height: triangles, parallelograms and trapeziums need the perpendicular height.
- Forgetting the ½ in the triangle and trapezium formulas.
- Stopping at \(x\): if the question asks for a side, area or perimeter, use \(x\) to finish the job.
- Keeping a negative root: a length cannot be negative — reject it and say why.
- Paper 2: one- or two-mark unit conversions (cm² ↔ m², cm³ ↔ litres, km/h ↔ m/s). These are quick marks if you square or cube the factor.
- Paper 4: areas with algebraic sides — “Show that \(x^{2} + 5x – 36 = 0\)” then “Solve” — and practical tanks, fields and costs, often 4–6 marks across several parts.
- Command words: “Calculate”, “Find”, “Show that” (every step must be written), “Give your answer in …” (check the unit asked for).
- Check the formula list at the start of the paper for area formulas, but know them well enough to use them fast. The unit conversion factors (10 000, 1 000 000, 1000 cm³ = 1 litre) are never given — learn those by heart.