Cross Section and Sketch Maps

Lessons

Senior 6 · Geography

Cross Section and Sketch Maps

4 lessons

Lesson 4 of 4

Enlarging and Reducing Maps

22 min

Same area, different size

On a 1:50 000 sheet a secondary school's whole campus is smaller than a fingernail. A district engineer planning a new classroom block needs that corner of the map drawn larger; a student fitting the same area into a report needs it smaller. Changing the size of a map changes its scale, and the examination tests both the drawing and the arithmetic: the new scale, the new size of the frame and what happens to areas.

Definition

Enlargement: redrawing a map larger than the original. Reduction: redrawing it smaller.

The linear scale factor $k$ compares lengths on the new map with the same lengths on the original:

k = \frac{\text{length on the new map}}{\text{length on the original map}}

$k > 1$ is an enlargement, $k < 1$ a reduction. Example: a 2 cm grid square redrawn as 3 cm gives $k = \frac{3}{2} = 1.5$ .

What happens to the scale and to areas

Enlarging a map by $k$ makes every length on it $k$ times longer, but the ground it shows stays the same. So each centimetre now stands for less ground, and the denominator of the representative fraction (RF) is divided by $k$ :

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

Enlarging 1:50 000 by 2 gives 1:25 000; reducing it by half ( $k = \frac{1}{2}$ ) gives 1:100 000.

Areas grow with the square of the linear factor, because both the length and the width are multiplied by $k$ :

\text{area on the new map} = k^2 \times \text{area on the original map}

Important

"Enlarge twice" means twice the length, which gives four times the area. To double the area of a map, the linear factor is $\sqrt{2} \approx 1.41$ .

Worked example 1: enlarging part of Map 1

Problem. Enlarge the part of Map 1 (Lesson 1) between eastings 14 and 16 and northings 21 and 24, which contains Hill Y, to twice its size. The original grid squares are 2 cm at 1:50 000. Give the frame size and the new scale.

Solution. The area is 2 squares wide and 3 squares high. With $k = 2$ , each square becomes $2 \times 2 = 4$ cm, so the frame is $2 \times 4 = 8$ cm wide and $3 \times 4 = 12$ cm high.

\text{new RF denominator} = \frac{50\,000}{2} = 25\,000 \quad\Rightarrow\quad 1:25\,000

On the new map 1 cm represents 250 m, so the redrawn bar scale shows 1 km as 4 cm.

Worked example 2: reducing to a given scale

Problem. A plan at 1:25 000 has 4 cm grid squares. It must be redrawn at 1:40 000. How large are the new squares?

Solution. Rearranging the rule, $k = \frac{\text{old denominator}}{\text{new denominator}}$ :

k = \frac{25\,000}{40\,000} = 0.625, \qquad \text{new square} = 0.625 \times 4\ \text{cm} = 2.5\ \text{cm}

Check: 2.5 cm at 1:40 000 is $2.5 \times 40\,000 = 100\,000$ cm = 1 km, and 4 cm at 1:25 000 is also 1 km. The squares still stand for the same ground.

Worked example 3: areas before and after

Problem. A forest covers 1.5 km² of ground. How much space does it take on the 1:50 000 map, and on the map enlarged 2 times?

Solution. At 1:50 000, 1 km is 2 cm, so 1 km² is $2 \times 2 = 4\ \text{cm}^2$ . The forest takes $1.5 \times 4 = 6\ \text{cm}^2$ . After enlarging by $k = 2$ it takes $2^2 \times 6 = 24\ \text{cm}^2$ . The ground area is still 1.5 km²: only the drawing has grown.

The grid-square method

Draw a grid over the original, or use its printed grid lines.
Work out the new square size: $k \times$ the old one. Draw a frame with the same number of squares and number the lines as on the original.
Mark every point where a feature crosses a grid line, in the matching place on the new grid (the red dots in Figure 8).
Join the marks, copying the shape of the feature inside each square.
Add the other features, then the title, key, north arrow and the new scale, drawn as a new bar scale.

Figure 8: enlarging by the grid-square method. The river is fixed by its four crossing points before it is drawn; the lake and village are placed by their position inside their squares.

The smaller the squares, the more crossing points each feature has, and the more accurate the copy. For a detailed feature, divide each grid square into four before you start.

Other methods

Method	How it works	Strength and weakness
Grid squares	Copy square by square onto a larger or smaller grid	Needs only a ruler and pencil; slow for detailed maps
Pantograph	Linked arms: a pointer traces the original while a pencil draws the copy	Fast and accurate once set; needs the instrument and a large table
Proportional dividers	Dividers with a sliding pivot; the two ends open in a fixed ratio	Transfers each distance at the new scale; one distance at a time
Photocopier or computer (GIS)	Enlarge or reduce by a percentage	Quick; the printed scale statement no longer fits and must be corrected, but a bar scale changes size with the map and stays correct

Worked example 4: fitting a map to the space

Problem. An extract at 1:25 000 measures 15 cm by 12 cm. It must fit a space 10 cm wide. Find the linear factor, the new height and the new scale.

Solution.

k = \frac{10}{15} = \frac{2}{3}, \qquad \text{height} = \frac{2}{3}\times 12 = 8\ \text{cm}, \qquad \text{new RF denominator} = \frac{25\,000}{\frac{2}{3}} = 37\,500

The new scale is 1:37 500. The frame must be 10 cm by 8 cm, not 10 cm by 12 cm: both sides change by the same factor, or the map is distorted.

Warning

Common mistakes

Multiplying the denominator when enlarging. Enlarging makes the denominator smaller: 1:50 000 enlarged 2 times is 1:25 000, not 1:100 000.
Scaling only one side. Change the width and height by the same factor.
Forgetting that area goes with $k^{2}$ .
Copying the old scale. A reduced map with the original "1:50 000" printed on it is wrong. Write the new RF and draw a new bar scale.

Tip

Check your new scale. Take one grid square: work out the ground distance it represents on the old map and on the new one. If they are not equal, the scale is wrong.

Guyana Learning Channel: "Geography – Grade 11: Enlarging, Copying and Reducing a Map". A classroom lesson on enlarging and reducing maps. As you watch, check each new scale against the rule new denominator = old denominator ÷ k.

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Practice

A 1:50 000 map is enlarged 2.5 times. What is the new scale?
A 1:20 000 map with 5 cm grid squares is reduced to 1:50 000. Find the linear factor and the new size of each square.
A map at 1:50 000 is to be redrawn with double the area. Find the linear factor and the new scale, to the nearest 100.
A lake covers 3 cm² on a 1:50 000 map. What is its area on the ground in km², and how much space will it take on a copy reduced to half size?
The bar scale of a map shows 1 km as 2 cm. How long is 1 km on the bar scale after the map is enlarged 1.5 times?
A pupil enlarges a 1:50 000 map twice and writes "new scale 1:100 000". Explain the error and give the correct scale.
Why does the grid-square method become more accurate when the squares are made smaller?

Show answersHide answers

$\frac{50\,000}{2.5} = 20\,000$ : 1:20 000.
$k = \frac{20\,000}{50\,000} = 0.4$ ; new square $0.4 \times 5 = 2$ cm.
$k = \sqrt{2} = 1.414$ ; $\frac{50\,000}{1.414} = 35\,355$ , so about 1:35 400.
At 1:50 000, 1 cm is 0.5 km, so 1 cm² is $0.5 \times 0.5 = 0.25\ \text{km}^2$ : ground area $3 \times 0.25 = 0.75\ \text{km}^2$ . On the half-size copy, $\left(\frac{1}{2}\right)^2 \times 3 = 0.75\ \text{cm}^2$ .
$1.5 \times 2 = 3$ cm.
Enlarging makes each centimetre stand for less ground, so the denominator is divided by 2, not multiplied: the correct scale is 1:25 000.
Each feature crosses more grid lines, so more of its points are fixed exactly and less of its shape has to be guessed between them.

Summary

Linear factor $k$ = new length ÷ old length; new RF denominator = old denominator ÷ $k$ .
Areas change by $k^{2}$ ; the ground shown does not change.
The grid-square method: new squares $k$ times the old, transfer crossing points first, then redraw the scale.

1Same area, different size

2What happens to the scale and to areas

3The grid-square method

4Other methods

Same area, different size

What happens to the scale and to areas

The grid-square method

Other methods