Chapter 1: Scales
Picture a trekker in the Western Himalayas, planning tomorrow’s walk to a high pass. She has one sheet of paper — a Survey of India topographical map — and a ruler. The trail she must follow twists along a valley, and on the map it is a thin, wandering line barely three centimetres long. Three centimetres. It looks like nothing. But she does not pack for three centimetres of walking.
At the bottom of the sheet is a small printed line: a scale. It tells her the map is drawn at 1:50,000 — that every one centimetre on the paper stands for fifty thousand centimetres, which is five hundred metres, on the mountain. She lays a thread along the winding trail, straightens it against the ruler, and reads eleven centimetres. Eleven centimetres, at five hundred metres each, is five and a half kilometres — uphill, in thin air. Now she knows to start at dawn.
This is the quiet power of a scale. A map is the whole world shrunk to fit in your hands, and the scale is the exchange rate between the two — the single figure that turns a centimetre of ink back into a kilometre of ground. Get it wrong, and a short line becomes a long day. Get it right, and a sheet of paper becomes a trustworthy guide across country you have never seen.
Diagram
Fig. 1.0-1 Caption: The scale links a small distance on the map to a large distance on the ground. Brief: Two panels side by side. Left panel, a small rectangle labelled ‘Map’, with two dots A and B joined by a short straight line marked ‘1 cm’. Right panel, a larger rectangle labelled ‘The ground’, with the same two points A and B joined by a longer line marked ‘500 m’. A curved arrow runs from the map line across to the ground line, pointing right, labelled ‘read through the scale’. Below both panels, print the scale in words: 1 cm : 500 m (1 : 50,000). No terrain, contours or shading — keep it a clean concept sketch.
1.1 What a Scale Is and Why a Map Needs One
The earth is vast, and a sheet of paper is small. No sheet is large enough to show a country, a district or even a whole town at its true size, and a map drawn that big would be impossible to handle. So every map is a reduced picture of the ground — the world shrunk down to fit within its margins.
The mapmaker must therefore decide, before drawing, how much real ground each small length on the map will stand for. This fixed relationship between a distance on the map and the matching distance on the ground is the scale of the map. It is the rule that lets us read real distances back off a flat sheet.
A scale is a ratio — a comparison of two lengths measured in the same way. It can be written as:
\[\text{Scale} = \frac{\text{Map distance between two points}}{\text{Ground distance between the same two points}}\]
Here map distance is the length you measure on the paper with a ruler, and ground distance is the true distance between those same two points on the earth. Both must be expressed in the same unit before they are compared.
Suppose two points, A and B, lie 1 cm apart on a map, while the real distance between them is 1 km. The scale is then 1 cm : 1 km. Every centimetre of this map, wherever you lay the ruler, stands for one kilometre of ground.
Why A Scale Matters
Why does a map need a scale at all? For two reasons. First, the scale is what lets us represent the earth, or any part of it, at whatever size the paper allows — a wall map and a pocket map of the same district differ only in their scale. Second, the scale is what lets us measure. A map without a scale is no more than a rough sketch: it may show that one place lies north of another, but never how far apart the two are.
You have already met scale bars in an atlas, printed as a small ruled line near the map’s edge. Everything in this chapter is the machinery behind that little line — what it means, how it is written, and how it is drawn.
1.2 The Three Ways of Showing a Scale
A scale can be printed on a map in three ways: as a statement of scale, as a representative fraction, or as a linear scale. All three say the same thing — how much ground each length on the map stands for — but they say it in different languages, and each has its own strength and its own weakness. A mapmaker chooses among them, and often prints two together.
1.2.1 Statement of Scale
The statement of scale expresses the scale in plain words. We simply state it: one centimetre to one kilometre, or one inch to one mile. This tells the reader directly that one centimetre on the map corresponds to one kilometre on the ground.
Its strength is that anyone can understand it at a glance, it takes no time to write, and it gives an immediate, honest sense of distance. Its weakness is twofold. It makes sense only to a reader who already knows the unit named — the words ‘one inch to one mile’ mean nothing to someone who thinks in kilometres. And if the map is later photographed larger or smaller, the words stay the same while the true relationship has changed, so the printed statement becomes wrong.
1.2.2 Representative Fraction
The representative fraction, written R.F. and also called the numerical fraction, states the scale as a fraction whose numerator is always 1. It is the ratio of map distance to ground distance, with both measured in the same unit:
\[\text{Representative Fraction (R.F.)} = \frac{\text{Map distance}}{\text{Ground distance}} \qquad (\text{both in the same unit})\]
Because the two distances share a unit, the unit cancels and the fraction has no unit of its own. An R.F. of 1 : 1,00,000 means one length on the map represents one lakh of the same lengths on the ground — one centimetre standing for 1,00,000 centimetres, or one inch for 1,00,000 inches.
This is the great strength of the R.F.: it belongs to no country’s measuring system. A reader in Britain reads it in inches, a reader in France in centimetres, and a reader in Russia in versts — the verst being an old Russian unit of distance — and all three read the same map correctly. For this reason the R.F. is also called the International Scale or the Natural Scale.
Its weakness is that you cannot measure a distance directly from a bare fraction; you must first convert it into a statement you can work with. And, like the statement of scale, the printed R.F. becomes untrue the moment the map is photographically enlarged or reduced, because the map distance changes while the fraction on the page does not.
1.2.3 Linear Scale
The linear scale, also called the graphical or plain scale, is a straight line drawn on the map itself, marked off so that distances can be read from it directly. It is divided into primary divisions, the main marked units that show whole distances, and, within the first primary division, into secondary divisions, smaller subdivisions that let you read shorter distances.
One feature of its layout surprises students at first. The zero is not placed at the extreme left of the line. It sits to the right of the secondary divisions, so that the subdivided first block lies to the left of zero and the whole primary divisions run to the right. To measure, you step the map distance off against the bar and read the ground distance straight from the labels — no arithmetic is needed.
We refer to the layout in the diagram below.
Diagram
Fig. 1.2-1 Caption: On a linear scale the zero sits to the right of the secondary divisions, with whole units to its right. Brief: Draw a horizontal linear (plain) scale as a long rectangle divided into equal blocks, shading alternate blocks black and white so lengths are easy to step off. Mark a clear zero point one primary division in from the left end. To the RIGHT of the zero, show four equal primary divisions with tick labels 10, 20, 30, 40 (km). To the LEFT of the zero — the first primary division — subdivide it into five equal secondary divisions with tick labels 2, 4, 6, 8, 10 (km) counting outward to the left. Label the right-hand portion ‘primary divisions’ and the left-hand subdivided portion ‘secondary divisions’. Add a small note with an arrow to the zero: ‘zero sits to the right of the secondary divisions’. Print an R.F. below the bar. This is a concept diagram, not a scale constructed to any one map.
The linear scale carries one advantage the other two lack. If the map is photographed larger or smaller, the printed bar is enlarged or reduced by exactly the same amount as the map. The bar therefore stays true to the map whatever its size — the property that makes it the safe scale to trust on a photocopy. Against this, it shares the statement’s limit that a reader must know the unit it is marked in, and drawing an accurate bar takes some care and time.
Which To Use When
The three methods survive together because each answers a different need. The statement of scale is the quickest to read and the friendliest to a general reader. The representative fraction is unit-free and works in any country, which is why it appears on maps meant to travel across borders. The linear scale is the one you can lay a distance against and read off at once, and the only one that stays honest when the map is resized.
A well-made map therefore often prints two scales side by side — the R.F. for universality, and a linear scale for direct, resize-proof measurement — so the reader gets both strengths at once.
1.3 Converting Between the Statement and the Representative Fraction
The statement of scale and the representative fraction carry the same information, so either can be rewritten as the other. This conversion is the most common piece of scale arithmetic, and it rests on two fixed facts you must keep to hand: one kilometre equals 1,00,000 centimetres, and one mile equals 63,360 inches.
1.3.1 Statement of Scale into Representative Fraction
To turn a statement into an R.F., start from the defining ratio and make the two distances share a single unit, because an R.F. carries no unit of its own:
\[\text{R.F.} = \frac{\text{Map distance}}{\text{Ground distance}} \qquad (\text{same unit, top and bottom})\]
Take a statement of 1 cm : 4 km. The map distance is already in centimetres, so convert the ground distance to centimetres as well:
\[4\text{ km} = 4 \times 1,00,000\text{ cm} = 4,00,000\text{ cm}\]
\[\text{R.F.} = \frac{1\text{ cm}}{4,00,000\text{ cm}} = \frac{1}{4,00,000} \quad\text{or}\quad 1 : 4,00,000\]
The same method works in British units. Take a statement of 1 inch : 2 miles. Convert the ground distance to inches:
\[2\text{ miles} = 2 \times 63,360\text{ inch} = 1,26,720\text{ inch}\]
\[\text{R.F.} = \frac{1\text{ inch}}{1,26,720\text{ inch}} = \frac{1}{1,26,720} \quad\text{or}\quad 1 : 1,26,720\]
Notice that the unit vanishes once numerator and denominator match. The single rule to remember is: never write an R.F. until both distances are in the same unit.
1.3.2 Representative Fraction into Statement of Scale
Going the other way, you expand the fraction into a statement and then reduce the large ground figure to a unit the reader can picture. Any unit works, but choose one the reader will find familiar.
Take an R.F. of 1 : 1,90,080. It means one inch on the map represents 1,90,080 inches on the ground. Inches are hard to imagine over that distance, so convert them to miles:
\[\frac{1,90,080\text{ inch}}{63,360\text{ inch per mile}} = 3\text{ miles}\]
The statement of scale is therefore 1 inch : 3 miles — far easier to read than the raw fraction.
The same expansion in metric units gives a real map its everyday meaning, as the case below shows.
What that figure means is the point of the exercise. On this sheet a single centimetre swallows two and a half kilometres of ground, so one printed sheet spans a wide region — but it cannot show fine detail, because a footpath or a small field is far thinner than the ink can carry. Compare it with the 1:50,000 toposheet from the start of this chapter, where 1 cm held only 500 m. The degree sheet has the larger denominator, so it is the small-scale map — one that packs more ground into each centimetre and shows less detail — while the toposheet, with more detail over a smaller area, is the large-scale map. Which one you reach for depends on whether you are planning a region or walking a valley.
1.4 Constructing a Graphical Scale
Knowing a scale is one thing; drawing one is another. A graphical, or plain, scale can be constructed whenever the R.F. or the statement of scale is given. The finished bar then lets a reader measure real distances straight off the map. Before the bar can be built, though, we need a way to divide a line into exactly equal parts.
1.4.1 Dividing a Straight Line into Equal Parts
A scale bar must be split into equal divisions, and a length such as 13.33 cm cannot be split into five equal parts by eye or by casual ruler marks. Geometry solves this exactly. Two standard methods are used, and both rest on the same fact: a set of parallel lines cuts equal intercepts on one line whenever it cuts equal intercepts on another crossing line.
In the first method, suppose the line XY is to be divided into five equal parts. From X, draw a second line XE at any convenient acute angle. Along XE, step off five equal lengths with a compass, marking points A, B, C, D and E. Join E to Y, then draw lines through A, B, C and D parallel to EY. Where these parallels meet XY, they cut it into five exactly equal parts.
Diagram
Fig. 1.4-1 Caption: The first method — parallels drawn from equal steps on a slanting line cut the base into equal parts. Brief: Draw a horizontal line XY. From X, draw a second line XE rising at an acute angle. Along XE, mark five equally spaced points A, B, C, D, E. Join E to Y with a solid line. Through A, B, C and D draw dashed lines parallel to EY, each meeting XY at points labelled 1, 2, 3, 4. Mark the five equal segments on XY as X-1, 1-2, 2-3, 3-4, 4-Y. Show the compass-equal steps on XE with small tick marks. Direction: construction runs from X outward.
The second method uses two parallel lines. From X draw XE and from Y draw YJ, the two lines parallel to each other on opposite sides of XY. Mark five equal points A to E along XE and five equal points F to J along YJ, keeping the spacing the same on both. Now join X to J, A to I, B to H, C to G, D to F and E to Y. These joins cross XY at four points that divide it into five equal parts.
Diagram
Fig. 1.4-2 Caption: The second method — crossing joins between two parallel lines divide the base evenly. Brief: Draw a horizontal line XY. From X draw XE rising and from Y draw YJ falling, the two lines parallel to each other. Along XE mark five equal points A, B, C, D, E; along YJ mark five equal points F, G, H, I, J at identical spacing. Draw straight joins X-J, A-I, B-H, C-G, D-F and E-Y; mark where each crosses XY as points 1, 2, 3, 4. Label the five equal segments X-1, 1-2, 2-3, 3-4, 4-Y.
1.4.2 Drawing a Plain Scale
With equal division in hand, a plain scale is built by following a short, fixed procedure. A few precautions keep the finished bar accurate and easy to read.
- Keep the line between 12 and 20 cm long (or 5 to 8 inches).
- Let it stand for a round ground distance — one divisible by 2, 5, 10 and so on — so the marks fall on convenient numbers.
- Divide the whole line into equal primary divisions, each a round distance.
- Subdivide only the first primary division into secondary divisions for shorter distances.
- Place the zero to the right of the secondary divisions, not at the far left.
- Mark primary distances to the right of zero and secondary distances to the left.
- Write the R.F. above the line, and the statement of scale too if you wish.
A Scale That Comes Out Round
Take a map with an R.F. of 1 : 2,00,000. Converting to a statement, 1 cm represents 2,00,000 cm, which is 2 km. So the scale is 1 cm : 2 km.
Choose a line 15 cm long, comfortably within the 12–20 cm range. At 2 km to the centimetre, this line stands for:
\[15\text{ cm} \times 2\text{ km per cm} = 30\text{ km}\]
Thirty is already round, so no adjustment is needed. Divide the 15 cm line into six equal primary divisions, each showing 5 km, and subdivide the first primary division into five secondary divisions of 1 km each. Set the zero one primary in from the left, and the scale is complete.
Diagram
Fig. 1.4-3 Caption: A plain scale for R.F. 1 : 2,00,000, reading up to 30 km. Brief: Draw a finished plain scale 15 cm long representing 30 km, as a horizontal bar with alternate blocks shaded black and white. Six equal primary divisions of 5 km. The leftmost primary division subdivided into five equal secondary divisions of 1 km. Place the zero at the junction, one primary division in from the left end. To the right of zero, primary tick labels 5, 10, 15, 20, 25 (Km). To the left of zero, secondary tick labels 1, 2, 3, 4, 5 (Km) counting outward. Print ‘R.F. 1 : 2,00,000’ above the bar and ‘Km’ at both ends.
When The Length Is Not Round
Most real R.F.s do not give a round distance for a neat line length, and then we work backwards from a round distance to the length that shows it.
The point of the adjustment is what the finished bar can now do. Because the line was measured out for a round 100 km and divided evenly, every mark falls on a convenient number — 20, 40, 60, 80 km — and the reader steps a map distance against the bar and reads the ground distance off at once, with no fraction to expand and no sum to do. That direct, arithmetic-free reading is the whole reason the graphical scale is worth the trouble of drawing.
Diagram
Fig. 1.4-4 Caption: A plain scale for R.F. 1 : 7,50,000, adjusted to read up to 100 km. Brief: Draw a finished plain scale 13.33 cm long representing 100 km, as a horizontal bar with alternate blocks shaded black and white. Five equal primary divisions of 20 km. The leftmost primary division subdivided into five equal secondary divisions of 4 km. Place the zero at the junction, one primary division in from the left. To the right of zero, primary tick labels 20, 40, 60, 80 (Km). To the left of zero, secondary tick labels 4, 8, 12, 16, 20 (Km) counting outward. Print ‘R.F. 1 : 7,50,000’ above the bar and ‘Km’ at both ends.
1.5 Summary
A map must be smaller than the ground it shows, because no paper could hold the earth at true size. The scale records exactly how much the ground has been reduced. It is the ratio of map distance to ground distance, with both measured in the same unit.
A scale is what makes a map more than a rough sketch. It lets the same district be drawn as a wall map or a pocket map, and it lets a reader measure real distances off the paper.
A scale can be shown in three ways. The statement of scale gives it in words; the representative fraction (R.F.) gives it as a unit-free fraction with numerator 1; the linear scale gives it as a drawn, divided bar.
Each method has a trade-off. The statement is quick to read but tied to one unit and breaks when the map is resized. The R.F. is universal and unit-free but not directly measurable, and it too breaks on resizing. The linear scale can be measured off directly and stays true when the map is enlarged or reduced.
To convert a statement into an R.F., put both distances in the same unit and write the ratio as a fraction with numerator 1, using the fixed facts that one kilometre is 1,00,000 centimetres and one mile is 63,360 inches.
To convert an R.F. into a statement, expand the fraction, then reduce the large ground figure to a unit the reader can picture, such as kilometres or miles.
The size of the R.F. tells you the kind of map. A bigger denominator means more ground packed into each centimetre — a small-scale map covering a wide area with little detail. A smaller denominator is a large-scale map, showing fine detail over a small area.
A plain scale is built by dividing a line into exactly equal parts by geometry, choosing a 12–20 cm line that shows a round ground distance, marking primary divisions for whole units and subdividing the first into secondary divisions, and placing the zero to the right of the secondary block.
Flowchart
Fig. 1.5-1 Caption: The whole chapter as one workflow — from two measured distances to a bar you can read off directly. Nodes: Map distance and ground distance (same unit) → Scale, the ratio between them → shown three ways → branch: statement of scale (in words); representative fraction (unit-free, numerator 1); linear scale (a divided bar) → statement and R.F. convert into each other by matching units (1 km = 1,00,000 cm; 1 mile = 63,360 inches) → linear scale is constructed: a 12–20 cm line for a round distance, divided into primary divisions with the first subdivided into secondary, zero set to the right of the secondary block → read the ground distance directly off the bar
What This Chapter Cannot Tell You
A scale tells you distance, and nothing else. It says how far apart two points are, but not in which direction they lie, nor how the land rises and falls between them.
A single scale is also never perfectly true everywhere on a map. Because a curved earth is drawn on flat paper, distances stretch or shrink across a sheet, and the printed scale holds exactly only along certain lines. On a small-scale map of a large region, that distortion can be large — a matter for the study of map projections, not for the scale bar alone.
1.6 Key Terms
- Large-scale map — A map with a small R.F. denominator, showing fine detail over a small area.
- Linear scale — A scale drawn as a divided straight line, so distances can be measured directly off it; also called the graphical or plain scale.
- Primary divisions — The main equal divisions of a linear scale, each marking a whole unit of ground distance.
- Representative fraction — The scale written as a unit-free fraction with numerator 1, giving map distance over ground distance in the same unit.
- Scale — The fixed ratio between a distance on the map and the matching distance on the ground.
- Secondary divisions — The smaller subdivisions of the first primary division of a linear scale, used to read shorter distances.
- Small-scale map — A map with a large R.F. denominator, covering a wide area with little detail.
- Statement of scale — The scale expressed in words, such as one centimetre to one kilometre.
1.7 Points to Ponder
A linear scale survives photocopying, while a statement and an R.F. do not. Yet most official maps still print the R.F. beside the bar. What is the mapmaker gaining that is worth trusting the reader with the more fragile scale?
A small-scale map hides the footpaths; a large-scale map hides the region. Is there any single ‘right’ scale for a map, or does the right scale depend entirely on the question being asked of it?
A scale bar claims to hold true across the whole sheet, yet a flat map of a curved earth cannot keep every distance honest. When you measure a distance off a scale, how much should you trust it, and where on the sheet would you trust it least?
The representative fraction works in any country because it drops all units. But a reader must still imagine what 2,50,000 centimetres feels like on the ground. Does a unit-free scale make a map easier to read, or only easier to share?
Two maps of the same town, one at 1:10,000 and one at 1:1,00,000, both fit on a single page. What has the smaller-scale map given up in order to cover ten times the ground — and could that very loss ever be an advantage?
1.8 Objective Questions
- The ratio between a distance on the map and the actual distance between the same points on the ground is called the [1 mark]
- legend
- scale
- grid
- projection
- In a representative fraction, the numerator is always [1 mark]
- 10
- 100
- 1
- equal to the denominator
- One kilometre is equal to [1 mark]
- 1,000 centimetres
- 10,000 centimetres
- 1,00,000 centimetres
- 10,00,000 centimetres
- Which method of showing scale remains true even after a map is photographically enlarged? [1 mark]
- statement of scale
- representative fraction
- linear scale
- all three equally
- Assertion (A): The representative fraction is called the International Scale. Reason (R): The R.F. has no unit of its own and can be read in any system of measurement. [1 mark]
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true, but R is false.
- A is false, but R is true.
- Match each item in Column A with its meaning in Column B. [4 marks]
| Column A | Column B |
|---|---|
| a. Statement of scale | i. a unit-free fraction with numerator 1 |
| b. Representative fraction | ii. covers a wide area with little detail |
| c. Primary division | iii. the scale expressed in words |
| d. Small-scale map | iv. the main whole-unit division of a linear scale |
Fill in the blank: One mile is equal to ______ inches. [1 mark]
State whether the following is true or false: A map without a scale can still be used to measure exact distances. [1 mark]
1.9 Check Your Concepts
Define scale, and state the two distances it compares. [2 marks]
Why does a representative fraction have no unit of its own? [2 marks]
Differentiate between the statement of scale and the representative fraction. [3 marks]
Give reasons why a linear scale is the safest scale to trust on a photocopied map. [3 marks]
Convert the following statements of scale into representative fractions, showing your working. Take 1 km = 1,00,000 cm and 1 mile = 63,360 inches. [4 marks]
- 1 cm : 6 km
- 1 inch : 4 miles
Explain the difference between a large-scale map and a small-scale map, and state one purpose each is best suited to. [4 marks]
Describe the three methods of showing a scale on a map, and state one advantage and one limitation of each. [6 marks]
1.10 Map, Diagram & Data Interpretation
The distance between two points on a map is 3 cm, while the actual distance between them on the ground is 12 km. Using R.F. = map distance ÷ ground distance, find the R.F. of the map and then express it as a statement of scale. Show every step. [4 marks]
A map has a representative fraction of 1 : 4,00,000. Construct a plain scale for it, at least 12 cm long, to read kilometres and showing both primary and secondary divisions. State the length of line you choose and the distance each primary division represents. [4 marks]
Study the linear scale in Fig. 1.2-1. Explain why the zero is placed to the right of the secondary divisions rather than at the extreme left, and describe how you would use the bar to measure a ground distance of 26 km. [3 marks]
The flowchart in Fig. 1.5-1 shows how a scale turns two measured distances into a bar you can read off directly. State one thing about the ground that this scale, however carefully drawn, still cannot tell you. [2 marks]