Reducing and Enlarging Maps

25 min

Let's begin

A sector officer needs Map A on a single page of a report, so it must shrink. A health worker planning home visits in Gasave needs its streets drawn large enough to write on. Both need the same map at a new size, and the new copy must still be measurable: distances and grid references must still work. This lesson shows how to reduce and enlarge a map by hand, and how to work out the scale and areas of the new copy.

Definition

Enlarging a map: redrawing it larger, so that every length is multiplied by the same linear factor $k > 1$ .

Reducing a map: redrawing it smaller, so that every length is multiplied by a factor $k < 1$ (for example $k = \frac{1}{2}$ ).

Representative fraction (RF): a scale written as a ratio, e.g. 1:50 000, meaning 1 unit on the map stands for 50 000 of the same units on the ground.

The grid-square method

Draw (or use) a grid of equal squares over the original map. Map A already has one: 1 km squares.
On clean paper, draw a grid with the same number of squares, each side multiplied by the factor: half as long to reduce by half, twice as long to enlarge twice.
Number and letter the squares the same way on both grids, so you always know which square you are copying.
Copy one square at a time. First mark where each line (river, road, boundary) cuts the sides of the square, then join the marks, following the shape inside the square.
Write the new scale: redraw the scale bar at the new size, or calculate the new RF. Add a title, a key and the north arrow.

Figure 6. The grid-square method: the same river copied onto squares twice as wide. Drawn by Ganzaa.

Important

The new scale. Multiplying every map length by $k$ divides the RF denominator by $k$ :

\text{new RF} = 1 : \frac{\text{old denominator}}{k}

Enlarge twice ( $k = 2$ ): 1:50 000 becomes 1:25 000. Reduce by half ( $k = \frac{1}{2}$ ): 1:50 000 becomes 1:100 000. A bigger map has a smaller denominator.

Worked example

Worked example 1: new scale. Map A (1:50 000) is enlarged so that each grid square becomes four times as wide. Find the new RF.

The linear factor is $k = 4$ .
New denominator: $\frac{50\,000}{4} = 12\,500$ .

Answer: 1:12 500. Figure 8 below is drawn at this scale: 1 cm on it stands for 125 m.

Figure 7. Map A reduced to half its size as a sketch map, scale 1:100 000: only the main features are copied. Drawn by Ganzaa.

Figure 8. Grid square 2341 (Gasave) enlarged four times, scale 1:12 500, with a tenths grid for six-figure grid references. Drawn by Ganzaa.

What happens to area

When lengths are multiplied by $k$ , areas on the map are multiplied by $k^{2}$ , because an area is a length times a length. The ground area does not change at all.

Copy of Map A	RF	Size on paper	Area on paper	Ground area
Original	1:50 000	12 cm × 10 cm	120 cm²	30 km²
Reduced by half	1:100 000	6 cm × 5 cm	30 cm²	30 km²
Enlarged twice	1:25 000	24 cm × 20 cm	480 cm²	30 km²

At 1:50 000, 1 cm stands for 0.5 km, so Map A's 6 km by 5 km area is 12 cm by 10 cm on paper.

Worked example

Worked example 2: area. The swamp on a copy of Map A enlarged twice covers 8 cm². (a) What area does it cover on the original map? (b) What is its real area?

Areas scale by $k^{2} = 2^{2} = 4$ , so on the original it covers $\frac{8}{4} = 2\,\text{cm}^2$ .
On the original (1:50 000), 1 cm stands for 0.5 km, so $1\,\text{cm}^2$ stands for $0.5 \times 0.5 = 0.25\,\text{km}^2$ .
Real area: $2 \times 0.25 = 0.5\,\text{km}^2$ .

Answer: (a) 2 cm²; (b) 0.5 km². Check with the enlarged copy: at 1:25 000, $1\,\text{cm}^2$ is $0.25 \times 0.25 = 0.0625\,\text{km}^2$ , and $8 \times 0.0625 = 0.5\,\text{km}^2$ .

Worked example

Worked example 3: finding the factor. A 1:50 000 extract is redrawn at 1:20 000. Is this an enlargement or a reduction, and by what factor?

$k = \frac{\text{old denominator}}{\text{new denominator}} = \frac{50\,000}{20\,000} = 2.5$ .
$k > 1$ , so the map gets bigger.

Answer: an enlargement, 2.5 times: a 4 cm road becomes 10 cm long.

Warning

The photocopier trap. If a map is photocopied or shown on a screen at a different size, a printed RF such as "1:50 000" becomes wrong, but a scale bar stays right, because it shrinks or grows with the map. That is why Map A on your screen should be measured with its scale bar or its 1 km grid squares, never with a ruler and the RF. When you redraw a map, always redraw the scale bar as well.

Guyana Learning Channel: a Grade 11 geography lesson on enlarging, copying and reducing a map with a grid. Watch how each square is copied separately and how the new scale is worked out.

Tip

A quick check. Enlarging makes the denominator smaller; reducing makes it bigger. If your "enlarged" map has a larger denominator, you have divided the wrong way.

Practice

Map A (1:50 000) is reduced to one third of its length. What is the new RF?
A 1:25 000 map is enlarged three times. What is the new RF?
A pupil copies Map A onto a grid with 1 cm squares instead of its 2 cm squares. Is this a reduction or an enlargement? Give the factor and the new RF.
On Map A reduced by half, how long is the scale-bar segment for 1 km?
A lake measures 3 cm² on a 1:50 000 map. Find its real area in km².
The same lake is drawn on a copy enlarged twice. What is its area on the copy?
Explain why the RF printed on a map is no longer correct after the map is photocopied at 70% of its size, but the scale bar still is.
In Figure 8, give the six-figure grid reference of the church, and the distance on the enlarged copy between the church and the health centre in centimetres.

Show answersHide answers

$k = \frac{1}{3}$ : new denominator $50\,000 \times 3 = 150\,000$ , so 1:150 000.
$\frac{25\,000}{3} \approx 8333$ , so about 1:8333.
A reduction to half ( $k = \frac{1}{2}$ ); new RF 1:100 000.
At 1:100 000, 1 km is $100\,000\,\text{cm} \div 100\,000 = 1\,\text{cm}$ long.
$1\,\text{cm}^2 = 0.25\,\text{km}^2$ , so $3 \times 0.25 = 0.75\,\text{km}^2$ .
$3 \times 2^2 = 12\,\text{cm}^2$ (the real area is still 0.75 km²).
Photocopying at 70% makes every length 0.7 times as long, so 1 cm on the copy no longer stands for 50 000 cm (the true RF becomes about 1:71 429). The scale bar is copied at 70% too, so it still shows the right length for 1 km.
Church: GR 234418. The health centre (GR 234416) is 0.2 km due south of it. At 1:12 500, 1 cm stands for 125 m, so 200 m is $\frac{200}{125} = 1.6\,\text{cm}$ .

1The grid-square method

2What happens to area

The grid-square method

What happens to area