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Topic

Measurement and Scale

Measurement and scale covers choosing units, converting units, using maps and plans, calculating size, estimating materials, and interpreting real measurements.

Topic explanation

Write units beside every value. Many errors happen because metres, centimetres, litres, and millilitres are mixed.

Scale drawings and maps connect a small drawing to a real object or place. The scale must be applied consistently.

Area uses square units such as m² and cm². Volume uses cubic units such as cm³ and m³. These units are part of the answer, not an optional extra.

Real-life material questions often need a sensible rounded answer. For example, paint or packs of flooring may need to be rounded up.

Tolerance gives an acceptable range. A measurement may be correct even when it is not exactly the stated value.

Quick methods

Conversions
Multiply to smaller units and divide to larger units.
Scale
Check whether the question asks for drawing length or real length before multiplying or dividing.
Area and cost
Find the area in square units first, then multiply by the cost per square metre.
Volume
Use length × width × height for cuboids, with cubic units such as cm³.
Compound shapes
Split the shape into rectangles, triangles, or other familiar parts.
Pythagoras and trigonometry
Draw or mark the right-angled triangle, then choose the side or angle relationship that matches the question.
Tolerance
Subtract and add the allowed variation.

Worked examples

Map scale

A map uses a scale of 1 cm = 2 km. Two towns are 7.5 cm apart on the map. Find the real distance.

  1. Multiply the map distance by the real distance for 1 cm.
  2. 7.5 × 2 = 15

Answer: The real distance is 15 km.

Watch out: Remember to convert the map answer into a real distance, so the unit is km.

Area and cost

A room is 4.2 m by 3.6 m. Carpet costs £12.50 per m². Estimate the total cost.

  1. Area = 4.2 × 3.6 = 15.12 m²
  2. Cost = 15.12 × £12.50 = £189.00

So: The carpet costs about £189.

Watch out: Find the area before calculating the cost. Area uses square units such as m².

Volume

A box is 40 cm by 30 cm by 20 cm. What is its capacity in litres?

  1. Volume = 40 × 30 × 20 = 24000 cm³
  2. 1000 cm³ = 1 litre
  3. 24000 divided by 1000 = 24

Final step: The capacity is 24 litres.

Watch out: Volume uses cubic units. Check that the intermediate answer is not left as cm or cm²

Paint coverage

A wall is 5 m wide and 2.4 m high. Paint covers 8 m² per litre. How many litres are needed for one coat?

  1. Area of wall = 5 × 2.4 = 12 m²
  2. Litres needed = 12 ÷ 8 = 1.5

Answer: 1.5 litres are needed. In real life, this may mean buying a 2-litre amount.

Watch out: Use the coverage rate after finding area. Avoid multiply the area by 8.

Ratio scale

A model car is built to a scale of 1:40. The real car is 4.8 m long. How long is the model in cm?

  1. Convert 4.8 m to 480 cm.
  2. A 1:40 model has length 480 ÷ 40 = 12 cm.

So: The model car is 12 cm long.

Watch out: Convert to matching units before dividing by the scale factor.

Right-angled triangle in context

A ladder reaches 3.6 m up a wall and its foot is 1.5 m from the wall. Estimate the ladder length.

  1. The wall, ground and ladder form a right-angled triangle.
  2. Use Pythagoras: ladder² = 3.6² + 1.5² = 12.96 + 2.25 = 15.21.
  3. Ladder length = √15.21 = 3.9 m

Final step: The ladder is about 3.9 m long.

Watch out: The ladder is the hypotenuse, so it should be the longest side.

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