Sec 2 Map Scale: Concepts & Worked Examples

Secondary 2 G3 Mathematics · Study guide

Convert map lengths, perimeters and areas, recover an unknown scale and compare maps with four worked examples and an interactive scale drawing.

Concept → worked examples → practice16 min read · practice linked below

A scale compares corresponding lengths

A scale of $1:50\,000$ means that one unit on the map represents 50 000 of the same unit on the ground. It could mean 1 cm to 50 000 cm, or 1 mm to 50 000 mm. The map and the real place have the same shape, with every length reduced by one common factor.

Use $1\text{ km}=1000\text{ m}=100\,000\text{ cm}$ . Hence $1:50\,000$ is also $1\text{ cm}:0.5\text{ km}$ . This form makes distance calculations easier.

For a scale written $1\text{ cm}:k\text{ km}$ , actual length in km = map length in cm × k. To work backwards, divide by k. Perimeter is a length, so it uses the same factor.

Convert the scale before using it

Worked example 1 — a road length and a reverse calculation

A map has scale $1:25\,000$ . A road measures 7.2 cm on the map. Find the actual length in kilometres, and the map length of a different 3.5 km road.

Convert the length represented by 1 cm: $25\,000\text{ cm}=250\text{ m}=0.25\text{ km}$ .
The 7.2 cm road represents $7.2\times0.25=\mathbf{1.8\text{ km}}$ .
For the 3.5 km road, each map centimetre accounts for 0.25 km. Its map length is $3.5/0.25=\mathbf{14\text{ cm}}$ .

Check: 4 cm represents 1 km. Thus 14 cm represents 3.5 km. State the requested unit in each answer; 1800 m is the same actual distance as 1.8 km.

Find an unknown scale

Worked example 2 — turn a measured pair into 1 : n

Two landmarks 4.8 km apart are 6 cm apart on a map. Find the scale as $1:n$ , then find the map perimeter of a park with actual perimeter 12 km.

Convert both distances to centimetres before simplifying the ratio:

$6:480\,000=1:80\,000.$

So $n=80\,000$ . Equivalently, 1 cm represents $4.8/6=0.8$ km. The park’s map perimeter is $12/0.8=\mathbf{15\text{ cm}}$ .

Check: multiplying 6 cm by 80 000 gives 480 000 cm, or 4.8 km. A perimeter is a sum of lengths, so the length factor applies to each part and to their sum.

A map with scale $1:40\,000$ shows the same place twice as large in length as one with scale $1:80\,000$ . The smaller denominator means that 1 cm covers less ground, allowing more detail.

Area uses the square of the length scale

If 1 cm represents k km, a 1 cm by 1 cm square represents a k km by k km square. The area factor is $k\times k=k^2$ . This applies to any region because its whole area is enlarged or reduced in two dimensions.

Compare the labelled dimensions. The drawings illustrate the relationship; their on-screen sizes are not physical map measurements.

Worked example 3 — convert map area and actual area

A map uses scale $1:50\,000$ . A reserve has map area 18 cm². Find its actual area. A second reserve has actual area 3.6 km²; find its map area.

The length scale is $1\text{ cm}:0.5\text{ km}$ .
Square both sides: $1\text{ cm}^2:0.25\text{ km}^2$ .
The first reserve has actual area $18(0.25)=\mathbf{4.5\text{ km}^2}$ .
Work backwards for the second: $3.6/0.25=\mathbf{14.4\text{ cm}^2}$ .

Check: one square kilometre takes four square centimetres on this map. So 3.6 km² takes $3.6\times4=14.4$ cm².

If you work entirely in centimetres, remember that $1\text{ km}^2=(100\,000)^2\text{ cm}^2=10^{10}\text{ cm}^2$ . A squared unit needs a squared conversion factor.

Work backwards from area to a length scale

Worked example 4 — compare two maps of the same lake

A lake of actual area 8 km² occupies 32 cm² on Map A and 8 cm² on Map B. Find each map’s scale in the form $1:n$ , and compare their map lengths.

For Map A, 1 cm² represents $8/32=0.25$ km². Take the positive square root to return to lengths: 1 cm represents $\sqrt{0.25}=0.5$ km = 50 000 cm. Scale A is $\mathbf{1:50\,000}$ .
For Map B, 1 cm² represents $8/8=1$ km². Hence 1 cm represents 1 km = 100 000 cm. Scale B is $\mathbf{1:100\,000}$ .
A real 3 km shoreline segment is $3/0.5=6$ cm on A and $3/1=3$ cm on B. Every corresponding length on A is twice its length on B.

Check: twice the length gives four times the area, and $32/8=4$ . The length ratio is $\sqrt4=2$ , not 4. All lengths are positive, so use the positive square root.

Explore a rectangular region

Change the map scale or the map dimensions

Keep the map dimensions fixed and change the scale to see how the represented ground area changes. Then double a map dimension and compare the length, perimeter and area.

Use the labelled centimetres. A screen can resize this diagram, so do not measure it with a ruler.

1 cm represents 0.5 km, so 1 cm² represents 0.25 km². Actual dimensions: 3 km by 2 km. Actual perimeter: 10 km. Actual area: 6 km².

Common mistakes and checks

Writing 1 : 0.5 for 1 cm : 0.5 km: ratio form requires matching units. The correct ratio is 1 : 50 000.
Multiplying an area by the length factor: write the area scale on a separate line before calculating.
Forgetting the square root in a reverse problem: an area ratio gives the square of the length ratio.
Using the wrong direction: with km per cm, multiply map centimetres to obtain actual kilometres; divide kilometres to get map centimetres.
Squaring a perimeter factor: perimeter measures length, even though it goes around an area.
Trusting an unscaled sketch: use stated measurements and scale, not screen size. Keep units with every result.

Practise with our topic questions

Use the Sec 2 map scale question bank to apply these methods. Attempt each question before opening its worked solution. Compare your reasoning as well as the final answer.