Map Projections, Class 11 Practical Work in Geography

What Map Projections is about

Key concepts

Map projection: meaning and need

NCERT section Need for Map Projection

Definition: the method of transferring the graticule of latitude and longitude onto a plane surface; the transformation of the spherical network of parallels and meridians onto a plane. The network of parallels and meridians is the graticule; drawing it on a flat surface is projection.

Globe versus map:

• A globe is the best model of the geoid-shaped earth: shapes, sizes, directions and distances are accurate.
• But it is expensive, hard to carry, and cannot show minor details; comparing two regions on it is also hard.
• On the globe meridians are semicircles and parallels circles; on a plane they become straight or curved intersecting lines.

Why distortion is unavoidable:

• A flat paper stuck on a globe will not fit over a large area without distortion.
• If light is thrown from the centre of the globe, the image is distorted away from the line or point of contact, and distortion increases with distance from the tangent point.
• The globe is not a developable surface, so tracing all properties (shape, size, direction) at once is nearly impossible.
• So projection is also defined as the study of methods tried for transferring the graticule to a flat sheet, each preserving some properties.

Worked example

Peel an orange in one piece and press the peel flat on a table. Use this to explain why every flat map of the world has some distortion. (2 marks)

Try it first, then show the solutionSolution

1. The peel tears or stretches when flattened because a sphere is a non-developable surface: it cannot be laid flat without shrinking, breaking or creasing.
2. The globe behaves the same way, so transferring its graticule to a plane always distorts shape, area, distance or direction, more so away from the point or line of contact; a projection can only keep the properties chosen.

Answer: A sphere, like the peel, cannot be flattened without tearing or stretching, so some distortion is unavoidable.

Elements of map projection

NCERT section Elements of Map Projection

1. Reduced earth: a model of the earth at a reduced scale on paper, roughly a spheroid (polar diameter shorter than equatorial), on which the graticule is transferred. In practice its radius is found as R = earth's radius ÷ scale denominator (NCERT uses 640,000,000 cm, or 250,000,000 inches).
2. Parallels of latitude: circles parallel to the equator, at uniform distance from the poles; each lies in a plane at right angles to the earth's axis; unequal in length (from a point at the poles to the full circumference at the equator); 0°-90° N and S.
3. Meridians of longitude: semicircles from pole to pole; two opposite meridians form a full circumference; all meet along the axis; the Greenwich meridian (0°) is the arbitrary reference.
4. Global properties to be preserved: (i) distance between points; (ii) shape of the region; (iii) size or area; (iv) direction of one point from another.

Worked example

Find the radius of the reduced earth for a map at 1:160,000,000, taking the earth's radius as 640,000,000 cm. (1 mark)

Try it first, then show the solutionSolution

1. R = 640,000,000 ÷ 160,000,000.
2. R = 4 cm.

Answer: 4 cm.

Classification of map projections

NCERT section Classification of Map Projections

Basis	Types	Key idea
Drawing technique	Perspective	Drawn with a source of light projecting the graticule onto a developable surface
	Non-perspective	Developed without light or shadows
	Conventional/mathematical	Derived by computation and formulae; little relation to a projected image
Developable surface	Cylindrical, conical, zenithal	A paper cylinder, cone or plane receives the graticule; a developable surface can be flattened, a non-developable one (the globe) cannot without shrinking, breaking or creasing
Position of contact	Normal (equatorial), oblique, polar	Touching at the equator, between pole and equator, or at the pole
Global property	Equal area (homolographic)	Areas correct
	Orthomorphic (true shape)	Shapes correct, usually at the cost of area
	Azimuthal (true bearing)	Directions from the centre correct
	Equidistant (true scale)	Scale correct only along selected parallels or meridians
Source of light	Gnomonic	Light at the centre of the globe
	Stereographic	Light at the periphery, diametrically opposite the point of contact
	Orthographic	Light at infinity, opposite the point of contact

No projection keeps all four properties at once; one is chosen for the property needed.

Worked example

A map-maker wants a world map on which areas are correct, while a sailor wants one on which straight lines keep a constant compass bearing. Name the global property each needs and say why one projection cannot satisfy both. (3 marks)

Try it first, then show the solutionSolution

1. Correct areas need an equal area (homolographic) projection.
2. Constant bearings with true shapes need an orthomorphic projection that keeps directions, such as Mercator's.
3. Keeping shapes and directions true stretches scale towards the poles, which enlarges areas, so no single projection keeps all global properties; each is chosen for the property needed.

Answer: Equal area (homolographic) for the map-maker; orthomorphic with true direction for the sailor; preserving one property sacrifices another.

Conical projection with one standard parallel: construction

NCERT section Constructing Some Selected Projections

Principle: the graticule is projected onto a cone touching the globe along one parallel, the standard parallel (usually the middle parallel of the area); other parallels are distorted (Figure 4.3).

NCERT example: area 10°N-70°N, 10°E-130°E, scale 1:250,000,000, interval 10°.

• R = 640,000,000 ÷ 250,000,000 = 2.56 cm.
• Standard parallel 40°N (middle of 10-70); central meridian 70°E (middle of 10-130).

Construction steps (NCERT, summarised):
1. Draw a quadrant of radius R; mark angles at the 10° interval (COE) and the standard parallel angle (40°) (BOE, AOD).
2. Draw tangents from A and B to meet at P: AP and BP are the sides of the cone touching the globe at the standard parallel.
3. The arc distance CE (10° of arc) is the spacing between parallels.
4. Drop XY perpendicular from OP to OB; XY gives the spacing of meridians along the standard parallel.
5. On a separate vertical line NS (the central meridian), mark the length BP from the pole: this is the radius of the standard parallel; draw it as an arc.
6. Mark other parallels at the arc distance CE along the central meridian and draw concentric arcs.
7. Mark XY repeatedly along the standard parallel for the meridians; join these points to the pole with straight lines.

Useful values (for R and standard parallel φ): parallel spacing = 2πR × interval ÷ 360; radius of the standard parallel on the cone = R × cot φ; meridian spacing on the standard parallel = 2πR cos φ × interval ÷ 360. For NCERT's example these give about 0.447 cm, 3.05 cm and 0.342 cm.

Worked example

For a conical projection with one standard parallel at 1:160,000,000 (R = 4 cm), standard parallel 40°N and 10° interval, find the spacing of parallels and the radius of the standard parallel. (3 marks)

Try it first, then show the solutionSolution

1. Parallel spacing = 2π × 4 × 10 ÷ 360 ≈ 0.698 cm.
2. Radius of the standard parallel = R × cot 40° = 4 × 1.192 ≈ 4.77 cm (the tangent length BP).
3. Other parallels: 30°N at 4.77 + 0.70 ≈ 5.47 cm and 50°N at 4.77 - 0.70 ≈ 4.07 cm from the cone's apex, drawn as concentric arcs.

Answer: Parallels about 0.70 cm apart; standard parallel radius about 4.77 cm.

Conical projection with one standard parallel: properties, limits, uses

NCERT section Constructing Some Selected Projections

Properties:
1. All parallels are arcs of concentric circles, equally spaced.
2. All meridians are straight lines meeting at the pole, crossing parallels at right angles.
3. Scale is true along all meridians (distances along meridians are accurate).
4. The pole is an arc of a circle.
5. Scale is true along the standard parallel but exaggerated away from it.
6. Meridians get closer towards the pole.
7. It is neither equal area nor orthomorphic.

Limitations:
1. Not suitable for a world map: extreme distortion in the hemisphere opposite the standard parallel.
2. Even within one hemisphere, unsuitable for large areas: distortion is large near the pole and the equator.

Uses:
1. Mid-latitude areas with limited latitudinal and larger longitudinal extent.
2. A long, narrow east-west strip parallel to the standard parallel is shown correctly.
3. Showing railways, roads, narrow river valleys and international boundaries along the standard parallel.
4. Examples: Canadian Pacific Railway, Trans-Siberian Railway, USA-Canada boundary, Narmada valley.

Worked example

Why is the conical projection with one standard parallel suitable for the Trans-Siberian Railway but not for a world map? (2 marks)

Try it first, then show the solutionSolution

1. The railway is a long, narrow east-west strip in mid-latitudes; along the standard parallel the scale is true, so such a strip is shown correctly.
2. For the world, distortion becomes extreme in the opposite hemisphere and near the pole and equator.

Answer: True scale along an east-west mid-latitude strip; extreme distortion for the whole world.

Mercator's projection

NCERT section Constructing Some Selected Projections

Origin: developed by the Dutch cartographer Gerardus Mercator (named in NCERT as Mercator Gerardus Karmer) in 1569; based on mathematical formulae; an orthomorphic (true-shape) projection.

Character: parallels and meridians are straight lines at right angles; spacing between parallels increases towards the poles; shows correct directions: a straight line between two points is a line of constant bearing, the loxodrome or rhumb line (Figure 4.6; great circles appear as curves).

NCERT construction (1:250,000,000, 15° interval):

• R = 250,000,000 ÷ 250,000,000 = 1 inch (earth's radius taken as 250,000,000 inches).
• Equator = 2πR = 6.28 inches; interval = 6.28 × 15 ÷ 360 = 0.26 inch.
• Draw the equator, divide into 24 equal parts, draw meridians at right angles.
• Mark parallels from the equator using the table (× R): 15° 0.265, 30° 0.549, 45° 0.881, 60° 1.317, 75° 2.027 (inches for R = 1).

Properties: (1) straight-line graticule at right angles; (2) all parallels equal to the equator; (3) all meridians equal and equally spaced but longer than on the globe; (4) parallel spacing increases poleward; (5) scale true only along the equator; other parallels too long (e.g. the 30° parallel is 1.154 times its globe length); (6) shape kept, but distorted at high latitudes; (7) small countries near the equator shown truly, exaggeration grows poleward; (8) azimuthal; (9) orthomorphic, as scale along the meridian equals scale along the parallel.

Limitations: great exaggeration in high latitudes: Greenland appears as large as the USA, though it is about 1/10th of it; the poles cannot be shown (the 90° parallel and meridians would be infinite).

Uses: world maps and atlas maps; navigation (sea and air routes); drainage patterns, ocean currents, temperature, winds and their directions, world rainfall and other weather elements.

Worked example

Draw (calculate) a Mercator graticule for the world at 1:320,000,000 with 15° interval, taking the earth's radius as 640,000,000 cm. Give the equator length, meridian interval and distances of 30° and 60° from the equator. (3 marks)

Try it first, then show the solutionSolution

1. R = 640,000,000 ÷ 320,000,000 = 2 cm; equator = 2π × 2 ≈ 12.57 cm.
2. Meridian interval = 12.57 × 15 ÷ 360 ≈ 0.52 cm (24 divisions).
3. Parallels: 30° at 0.549 × 2 ≈ 1.10 cm and 60° at 1.317 × 2 ≈ 2.63 cm from the equator (using NCERT's table × R).

Answer: Equator 12.57 cm; 0.52 cm meridian spacing; 30° at 1.10 cm, 60° at 2.63 cm.

Map Projections in real life

Each concept, met in an everyday situation. Read the situation, answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Topics in this chapter

• Conical Projection with One Standard Parallel and Mercator's Projection: Properties, Construction and Uses
• Map Projections: Need, Distortion, Elements of a Projection and Classification

Map Projections: key facts

Reduced earth radius

$R=640,000,000cm÷scaledenominator(or250,000,000in÷denominator)$

1:250,000,000 → 2.56 cm or 1 inch.

Arc for an interval

$Arc=2\pi R\times interval÷360$

Parallel spacing on the conical projection; meridian spacing on Mercator's equator.

Conical: standard parallel radius

$Radiusoncone=R\times cot\phi (tangentlengthBP)$

φ = standard parallel.

Conical: meridian spacing

$Alongstandardparallel=2\pi Rcos\phi \times interval÷360(XY)$

Meridians are straight lines to the pole.

Mercator parallels

$Distancefromequator=tablevalue\times R:15°0.265,30°0.549,45°0.881,60°1.317,75°2.027$

Spacing increases poleward.

Mercator equator

$Equator=2\pi R;divideinto360÷intervalparts$

15° → 24 parts.

Mercator exaggeration

$30°parallel=1.154\times itsglobelength$

Poles cannot be shown.

Light source

Gnomonic	centre
stereographic	opposite periphery
orthographic	infinity

Classification by source of light.

Map Projections: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

Tricks to remember Map Projections

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Four classifications

  'Draw, Develop, Global, Glow': drawing technique, developable surface, global property, source of light.

  Use it when: Bases of classification.

• Light sources

  'Gnome in the Centre, Stereo on the Side, Ortho Out at infinity'.

  Use it when: Gnomonic, stereographic, orthographic.

• Global properties

  'DSAD': Distance, Shape, Area, Direction.

  Use it when: Properties to preserve.

• Mercator table

  '265-549-881-1317-2027' for 15°-75°.

  Use it when: Mercator construction.

Where students lose marks in Map Projections

The slips examiners see most often, and what to write instead.

• ✗Mistake: Drawing meridians as curves in the conical projection

  Confusion with the globe.

  Wrong working

  Curved meridians converging at the pole.

  Right working

  Meridians are straight lines meeting at the pole; parallels are concentric arcs.

  ✓Do this: Cone: straight meridians, curved parallels.

  Usually costs: Marks in construction.

• ✗Mistake: Spacing Mercator parallels equally

  Treating it like a simple cylindrical grid.

  Wrong working

  Parallels drawn at equal intervals.

  Right working

  Spacing increases towards the poles (0.265, 0.549, 0.881, 1.317, 2.027 × R).

  ✓Do this: Use the table, not a ruler step.

  Usually costs: Marks in construction.

• ✗Mistake: Calling Mercator's projection equal area

  Students remember 'world map'.

  Wrong working

  Mercator shows correct areas.

  Right working

  It is orthomorphic (true shape); areas are greatly exaggerated near the poles (Greenland vs USA).

  ✓Do this: Mercator = shape and direction, not area.

  Usually costs: Marks in property questions.

• ✗Mistake: Choosing the standard parallel at the edge of the area

  Not reading the data.

  Wrong working

  For 10°N-70°N, standard parallel 10°N.

  Right working

  Choose the middle parallel (40°N) to spread distortion evenly; central meridian in the middle of the longitudes.

  ✓Do this: Middle latitude, middle longitude.

  Usually costs: Marks in calculation.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What is a graticule?Show answer

   The network of parallels and meridians.

2. 2.What is a developable surface?Show answer

   One that can be flattened, such as a cylinder, cone or plane.

3. 3.What is another name for an equal area projection?Show answer

   Homolographic projection.

4. 4.Where does the cone touch the globe in a conical projection with one standard parallel?Show answer

   Along one parallel, the standard parallel.

5. 5.When did Mercator develop his projection?Show answer

   1569.

6. 6.What is a rhumb line?Show answer

   A straight line on Mercator's projection joining two points with a constant bearing (loxodrome).

7. 7.How much longer is the 30° parallel on Mercator than on the globe?Show answer

   1.154 times.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Practical exam chapter: 5 marks; only conical (one standard parallel) and Mercator in 2026-27.
• ☐Projection = transferring the graticule to a plane; globe is non-developable, so distortion is inevitable.
• ☐Elements: reduced earth, parallels, meridians, global properties (distance, shape, area, direction).
• ☐Classification: drawing technique; developable surface (cylindrical, conical, zenithal; normal, oblique, polar); global property; light source.
• ☐R = 640,000,000 cm ÷ scale denominator.
• ☐Conical: parallels concentric equally spaced arcs; straight meridians; true along meridians and the standard parallel; mid-latitude E-W strips.
• ☐Mercator (1569): orthomorphic, rhumb lines, parallels spaced 0.265-2.027 × R; navigation and world maps; Greenland exaggerated; poles not shown.

Map Projections: frequently asked questions