Units, Scale Drawings and Maps

Related unit conversions

Quantity	Useful facts
Length	1 m = 100 cm; 1 km = 1000 m
Mass	1 kg = 1000 g
Capacity	1 L = 1000 ml
Time	1 hour = 60 min; 1 min = 60 s

Choose one unit and convert every quantity before comparing or using a ratio.

Also use 1 cm = 10 mm, 1 tonne = 1000 kg, 1 day = 24 hours and 1 week = 7 days. Multiply when changing to smaller units and divide in the reverse direction: 4500 g = 4.5 kg. For time, 1 h 45 min = 1 + 45/60 = 1.75 h; 2.4 h = 2 h + 0.4 × 60 min = 2 h 24 min. Choose units suited to the size: mm for a screw, km for a road journey and litres for a bottle. Use any supplied conversion for other units.

Area conversions

Because 1 m = 100 cm, a 1 m by 1 m square is 100 cm by 100 cm. Therefore 1 m² = 10,000 cm².

The linear conversion factor is squared for area.

Worked example — convert area both ways

Convert 3.2 m² to cm². Since 1 m² = 10,000 cm², 3.2 × 10,000 = 32,000 cm².

Now convert 45,000 cm² to m². Divide by 10,000: 45,000 ÷ 10,000 = 4.5 m².

The direction matters: moving from larger square units to smaller square units multiplies by the squared conversion factor; reversing the conversion divides.

Volume conversions

Because 1 m = 100 cm, 1 m³ = 100³ cm³ = 1,000,000 cm³. The linear factor is cubed for volume.

Worked example — volume and capacity

Since 1 cm³ = 1 ml, 1000 cm³ = 1 L and 1 m³ = 1000 L. A volume of 0.004 m³ is 0.004 × 1,000,000 = 4000 cm³ = 4 L. Conversely, 750 ml = 750 cm³ = 0.00075 m³. Cube the length conversion factor before multiplying or dividing the volume.

Map scales

A scale of 1:25,000 means 1 unit on the map represents 25,000 of the same unit in reality.

If 6 cm is measured on a 1:25,000 map, the actual distance is 6 × 25,000 = 150,000 cm = 1.5 km.

Worked example — reverse a map scale

A village is 3.6 km from a station. The map scale is 1:20,000. How far apart should they be on the map?

Convert 3.6 km to centimetres: 3.6 km = 360,000 cm.

Map distance = 360,000 ÷ 20,000 = 18 cm.

For reverse scale questions, convert the real distance into the same units used by the scale before dividing.

Scale drawings

For a scale of 1 cm : 2 m, a real length of 7 m is drawn as 7 ÷ 2 = 3.5 cm. Use a ruler carefully and label the scale.

For a rectangular room measuring 8 m by 6 m, use 1 cm : 2 m to draw a 4 cm by 3 cm rectangle. Measure with a ruler and keep its corners at 90°. This is the same scale as 1:200 because 2 m = 200 cm. Drawings on a screen may be resized; construct the stated dimensions on paper. For bearings use Bearings and Geometric Drawings; for transformation centres use Enlargements and Centres of Enlargement.

Comparing lengths, areas and volumes

Compare supplied measures as ratios after matching units: 2 m² : 5000 cm² = 20,000:5000 = 4:1; 3 L : 750 ml = 3000:750 = 4:1. Unit conversion changes how the same measurement is expressed. Scaling a shape changes its size; the separate lesson Similar Shapes: Length, Area and Volume teaches the Higher area and volume scale-factor rules.

Think Like a Mathematician

Why does area use the square of the conversion factor?

Because both independent lengths are scaled. If each length is multiplied by 100, area is multiplied by 100 × 100.

Common Mistakes

Using 100 instead of 10,000: m² to cm² requires squaring the length factor.

Mixing map and real units: the ratio uses matching units first.

Confusing scale drawings with enlargement theory: this lesson is about numerical scale, not transformation centres.

Practice — units and scale

1. Convert 3.6 km to metres.

2. Convert 2.4 kg to grams.

3. Convert 1.8 m² to cm².

4. Convert 0.03 m³ to cm³.

5. A map scale is 1:50,000. A route measures 8 cm. Find the real distance in km.

6. On the same map, a real distance is 7.5 km. How many cm on the map?

7. A plan uses 1 cm : 0.5 m. Drawn length is 12 cm. Find the real length.

8. A 9 m wall is drawn at 1 cm : 1.5 m. Find the drawing length.

9. Simplify the ratio 2.5 m : 75 cm.

10. Convert 2500 cm³ to litres, and 2 hours 24 minutes to hours.

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