A Great Circle is a circle on the surface of the Earth whose centre and radius are those of the Earth itself.
It is called "great" because a disc cut in the plane of the Great Circle would have the largest area achievable — it divides the Earth into two equal hemispheres.
Figure 1 — A Great Circle — its centre and radius are those of the Earth itself
Key properties:
Shortest distance — the shorter arc of the Great Circle joining two points is the shortest path between them on the Earth's surface.
Uniqueness — given two non-antipodal points, only one Great Circle joins them.
Changing direction — a Great Circle track changes direction continuously relative to True North (except along a meridian or the Equator).
2. The Rhumb Line — Definition & Properties
🔎 Definition
A Rhumb Line is a regularly curved line on the surface of the Earth that cuts all meridians at the same angle — a line of constant direction.
Figure 2 — Moscow to Vancouver — Rhumb Line track plotted on a Mercator chart
The constant-direction property was enormously important to mariners and early aviators, who navigated by compass heading along straight Mercator chart tracks.
Unlike the Great Circle, the Rhumb Line is not the shortest distance (except along the Equator, a meridian, or a parallel of latitude).
Figure 3 — Moscow to Vancouver — Great Circle track; note how it swings close to the North PoleFigure 4 — Moscow to Vancouver — Great Circle track (curved, shorter) compared with Rhumb Line (straight, longer)
The Great Circle is always the shorter path; the Rhumb Line is always the longer one (again, except for the special cases — Equator, meridians).
Figure 5 — Great Circle track from Moscow to Vancouver plotted on a Mercator chart — track direction changes from ~330° to ~270° to ~210°
GC vs Rhumb Line — Relative Position
⚠ Critical Exam Rule
The Great Circle between any two points always lies nearer to the nearer Pole than the Rhumb Line.
The Rhumb Line always lies nearer to the Equator than the Great Circle.
In some textbooks: the Rhumb Line is described as convex to the Equator or concave to the nearer Pole.
There is only one Rhumb Line between any two points.
3. Lines That Are Both Great Circles and Rhumb Lines
✈ The Only Two Types
The only lines that are simultaneously Great Circles AND Rhumb Lines are:
The Equator — cuts all meridians at 90° (Rhumb Line property) AND its plane passes through the Earth's centre (Great Circle property).
Any Meridian (plus its anti-meridian) — a line of constant direction (0° or 180°) AND a semi-Great Circle.
Note: Parallels of latitude (other than the Equator) are Rhumb Lines but are Small Circles — their plane does not pass through the Earth's centre.
4. Great Circle Direction — The DIID Rule
Because a Great Circle track curves toward the nearer Pole, its direction changes continuously. The pattern of this change is captured in the DIID rule.
Figure 6 — Great Circle track direction change for an East–West Rhumb Line in the Northern and Southern hemispheres
Understanding DIID
Consider an East–West Rhumb Line track (090° or 270°) at ~50°N or ~50°S:
Hemisphere
Going East (→)
Going West (←)
Northern
GC track starts high (030°), swings to 090°, finishes low (150°) — track angle INCREASING
GC track starts low (330°), swings to 270°, finishes high (210°) — track angle DECREASING
Southern
GC track starts low (150°), swings to 090°, finishes high (030°) — track angle DECREASING
GC track starts high (210°), swings to 270°, finishes low (330°) — track angle INCREASING
Figure 7 — The DIID diagram — Decreasing, Increasing, Increasing, Decreasing — GC track direction change relative to the Rhumb Line
📚 Reading the DIID Diagram
Reading the four quadrants clockwise from the top-left: D–I–I–D (Decreasing–Increasing–Increasing–Decreasing).
North + East = Increasing
North + West = Decreasing
South + East = Decreasing
South + West = Increasing
In ALL cases: the GC direction always changes towards the Equator.
5. Distance on the Earth — Units & Conversions
Metric Units
Unit
Equivalent
1 metre (m)
100 cm = 1 000 mm
1 kilometre (km)
1/10 000th of Equator-to-Pole distance → Earth circumference = 40 000 km
1 km
3 280 ft
1 m
3.28 ft
Imperial Units
Unit
Equivalent
1 foot (ft)
12 inches
1 yard (yd)
3 feet
1 inch
2.54 cm
1 statute mile
5 280 ft
The Nautical Mile
✈ The Most Important Aviation Distance Unit
The nautical mile is directly related to the Earth's angular measurements:
1 NM = 1 minute of arc on a Great Circle
ICAO definition: 1 NM = 1 852 metres
Standard (Admiralty) NM: 1 NM = 6 080 ft
1 minute of latitude = 1 NM (everywhere, since all meridians are Great Circles)
1 minute of longitude = 1 NM only at the Equator
Angular Measurement
Distance
1° of latitude
60 NM
Equator to either Pole (90°)
5 400 NM
Earth circumference (360°)
21 600 NM
6. Variations in the Length of a Nautical Mile
Figure 8 — Geodetic and geocentric latitude — how the radius of curvature varies with latitude, causing the NM length to vary
The full definition: the NM is the arc of a Great Circle that subtends 1 minute of arc at the centre of curvature.
Because the Earth is an oblate spheroid, the radius of curvature varies with latitude:
Location
Radius of curvature
NM length
Equator
Smallest (most curved)
~6 048 ft (shortest)
Standard (mean)
Mean
6 076.1 ft (International NM = 1 852 m)
Poles
Largest (flattest)
~6 108 ft (longest)
📚 Exam Use
For all DGCA exam calculations, use the Standard NM of 6 080 ft.
The variation between 6 048 ft and 6 108 ft is not examined computationally — it is a conceptual understanding question only.
7. Conversion Factors
Conversion
Value
5 400 NM =
10 000 km
21 600 NM =
40 000 km
1 NM =
1 852 m = 6 080 ft
1 km =
3 280 ft
1 km =
0.5400 NM (= 5400/10000)
1 NM =
1.852 km
📚 Ratio Method for NM ↔ km
Use the circumference ratio: 21 600 NM = 40 000 km.
Therefore: NM × (40 000/21 600) = km | km × (21 600/40 000) = NM
Or simply: NM × 1.852 = km | km ÷ 1.852 = NM
8. Great Circle Distances — Six Worked Examples
🔎 Scope of DGCA Exam Questions
General GC distance calculation requires spherical geometry — not in the DGCA syllabus.
Exam questions are limited to points on special Great Circles: the same meridian, meridian & anti-meridian, or the Equator.
There are five general cases plus a special antipodal case — all six are shown below.
How to Identify Which Case Applies
Condition
Case
Same longitude, same hemisphere
Case 1 — Ch.lat direct
Same longitude, different hemispheres
Case 2 — Ch.lat = sum of lats
Longitudes add to 180°, same hemisphere
Case 3 — Route via nearer Pole
Longitudes add to 180°, different hemispheres
Case 4 — Route via nearer Pole
Both at 0° latitude (Equator)
Case 5 — Ch.long at Equator
Same lat N & S, lons add to 180°
Case 6 — Antipodal (10 800 NM)
Case 1 — Same Meridian, Same Hemisphere
Figure 9 — Case 1 — Same meridian, same hemisphere: London (51°37'N) to Accra (06°48'N) along 000°12'W
Example: London (51°37'N 000°12'W) to Accra (06°48'N 000°12'W)
Step 1 — Confirm same meridian: both at 000°12'W ✓
Case 3 — Meridian and Anti-Meridian, Same Hemisphere
Figure 11 — Case 3 — Meridian and anti-meridian, same hemisphere: Rome (41°55'N 011°10'E) to Honolulu (21°17'N 168°50'W) — route goes over the North Pole
Example: Rome (41°55'N 011°10'E) to Honolulu (21°17'N 168°50'W)
Direction of flight: Initially North (360°T), then South (180°T) after the pole
Case 4 — Meridian and Anti-Meridian, Different Hemispheres
Figure 12 — Case 4 — Meridian and anti-meridian, different hemispheres: Tokyo (35°57'N 135°35'E) to Rio de Janeiro (22°10'S 044°25'W) — route via the North Pole
Example: Tokyo (35°57'N 135°35'E) to Rio de Janeiro (22°10'S 044°25'W)
Step 2 — Route: Draw diagram — route via North Pole is shorter
Step 3 — Angular distance (via N Pole):
Tokyo → N Pole: 90° − 35°57' = 54°03'
N Pole → Equator: 90°
Equator → Rio: 22°10'
Total: 54°03' + 90° + 22°10' = 166°13'
Step 4 — NM: (166 × 60) + 13 = 9 973 NM
Check (via S Pole would give 193°47'): 360° − 193°47' = 166°13' ✓ — or: 21 600 − 11 627 = 9 973 NM ✓
✎ Self-Check: Wrong-direction check for Case 4
If your angular distance exceeds 180°, you've gone the wrong way around. Subtract from 360° for the angular answer, or subtract from 21 600 NM for the distance.
Case 5 — Two Points on the Equator
Figure 13 — Case 5 — Two points on the Equator: Dakar (000°N 016°35'W) to Singapore (000°N 103°55'E) — diagram viewed from above North Pole
Example: Dakar (00°00'N 016°35'W) to Singapore (00°00'N 103°55'E)
Step 1 — Both on Equator: ✓ (Equator is a Great Circle)
Step 2 — Ch.long: 016°35'W + 103°55'E = 120°30'E (going east from Dakar)
The conversion 1 minute of longitude = 1 NM is valid ONLY AT THE EQUATOR. At any other latitude, you must use the departure formula (covered in the Departure chapter).
Case 6 — Antipodal Points (Special Case)
Figure 14 — Case 6 — Antipodal points: Greenwich (51°30'N 000°00'E) and Antipodes Island (51°30'S 180°00'E/W) — diametrically opposite on the Earth
Example: Greenwich (51°30'N 000°00'E) to Antipodes Island (51°30'S 180°00'E)
Step 1 — Antipodal check: Equal latitudes N & S, longitudes add to 180° ✓
Step 2 — GC distance between any point and its antipode: always 180° = 10 800 NM
Note: Because the points are diametrically opposite, infinitely many Great Circles connect them.
9. Mean Latitude
The mean latitude between two positions is their arithmetic average. It is used in departure calculations (Ch.15).
Method
Same hemisphere: add both latitudes, divide by 2.
Different hemispheres: find the angular span (sum), halve it, subtract from the higher latitude.
✎ Worked Examples
Example 1 — same hemisphere: Mean lat of 52°17'N and 17°57'N
52°17' + 17°57' = 70°14' ÷ 2 = 35°07'N
Example 2 — different hemispheres: Mean lat of 35°25'N and 13°38'S
Total span = 35°25' + 13°38' = 49°03'. Half = 24°31.5'.
Mean lat = 35°25'N − 24°31.5' = 10°53.5'N
📚 Quick Revision Summary — Chapter 2
GC = shortest distance; constant direction = Rhumb Line
Rhumb Line is always nearer the Equator than the GC
GC always nearer the Pole than the Rhumb Line
Only lines that are both GC and Rhumb Line: Equator and meridians
DIID: North East = Increasing, North West = Decreasing, South East = Decreasing, South West = Increasing
GC direction always changes towards the Equator
1 NM = 1 852 m = 6 080 ft (standard); Earth = 21 600 NM = 40 000 km
NM length: shortest at Equator (6 048 ft), longest at Poles (6 108 ft)
Meridian+anti-meridian condition: longitudes of opposite signs that sum to 180°
Case 3/4: GC always goes via the nearer Pole
Antipodal: 10 800 NM (always)
Practice Questions & Detailed Answers
11 questions (many with sub-parts) • Calculation, conceptual & MCQ • Full worked solutions
Q1. What is the change of latitude between the following positions?
(a) 52°15'N to 39°35'N (b) 49°35'N to 60°20'S (c) 74°20'S to 34°30'S (d) 71°20'N to 86°45'N over the North Pole
Part (b) note: When crossing the Equator, add both latitudes (49°35' + 60°20' = 109°55').
Part (c) note: Both in Southern hemisphere — the aircraft travels northward (toward less southerly latitude). Subtract the smaller from the larger: 74°20' − 34°30' = 39°50'N.
Part (d) note: "Over the North Pole" means the route passes through 90°N and continues on the opposite meridian. From 71°20'N to the pole = 18°40'; pole to 86°45'N = 3°15'. Total = 21°55'.
Instructor's Note: Source key verified. All four answers confirmed. No discrepancy.
Q2. What is the distance in nautical miles and kilometres from position A (41°25'N) to position B (79°30'N)? Both are on the same meridian.
Step 3 — km: 2 285 × 1.852 = 4 232 km (or 4 230 km using the nav computer)
Instructor's Note: Source key: 2 285 NM / 4 232 km. Verified. No discrepancy.
Q3. What is the change of longitude between the following positions?
(a) 075°40'W to 125°35'W (b) 001°20'E to 004°20'W (c) 150°40'E to 179°30'E (d) 162°36'W to 140°42'E
Rule: same side → subtract; opposite sides → add. If the result exceeds 180°, take the supplement (360° − result) — this is the shorter arc.
Part (d) detailed: 162°36'W and 140°42'E are on opposite sides, so add: 162°36' + 140°42' = 303°18'. Since 303°18' > 180°, the shorter arc is 360° − 303°18' = 56°42'. Going westward from 162°36'W by 56°42' takes you through the anti-meridian to 140°42'E.
Instructor's Note: Source key notes "(d): Not 303°18' — we want the smaller arc." Confirmed. No discrepancy.
Q4. Give the direction and the change of latitude and longitude from X to Y in each case:
(a) 50°31'N 006°30'W → 52°00'N 008°35'W (b) 47°32'N 002°46'W → 43°56'N 001°33'W (c) 61°47'N 003°46'W → 62°13'N 001°36'E (d) 31°27'S 091°47'E → 35°57'N 096°31'E (e) 51°05'N 177°42'E → 51°06'N 167°42'W
(e) Lat: 51°06' − 51°05' = 00°01'N. Long: 177°42'E → 167°42'W — going east: 177°42'E to 180° = 2°18', then 180° to 167°42'W = 12°18'. Total = 014°36'E (shorter arc eastward).
Instructor's Note: All five sub-answers verified against source key. No discrepancies.
Q5. Give the shortest distance in nautical miles and kilometres between the following positions:
(a) 52°06'N 002°32'E and 53°36'N 002°32'E (b) 04°41'S 163°36'W and 03°21'N 163°36'W (c) 62°00'N 093°00'E and 62°00'N 087°00'W (d) 00°00'N 176°00'E and 00°00'N 173°00'W (e) 43°57'N 071°37'W and 43°57'S 108°23'E
Instructor's Note: All five answers confirmed against source key. No discrepancies.
Q6. An aircraft is to fly from 72°00'N 002°30'E to 72°00'N 177°30'W on the shortest possible route.
(a) Give the initial True track direction. (b) Will the track direction remain the same? (c) Why/why not?
(a) Initial track: 360°(T) — True North
(b) Track changes: No
(c) Route is over the North Pole. Initial track is True North; once past the pole, the track becomes True South (180°T).
Step 1 — Identify case: 002°30'E + 177°30'W = 180° → meridian/anti-meridian. Both at 72°N → same hemisphere → Case 3 (route over North Pole).
Step 2 — Initial direction: From 72°N heading to the North Pole = due North = 360°(T).
Step 3 — Track consistency: As the aircraft crosses the pole, "north" flips to "south" on the Mercator representation. The track direction changes from 360°T to 180°T — so No, it does not remain constant.
Instructor's Note: Verified against source key. No discrepancy.
Q7. You are at position A: 54°20'N 002°30'W. Given a ch.lat of 16°20'N and a ch.long of 20°30'W to B, what is the position of B?
Instructor's Note: Verified. No discrepancy. The longitude crosses the anti-meridian — a common exam trap.
Q9. What is the position of the Rhumb Line between 2 points relative to the Great Circle if the points are:
(a) In the Northern hemisphere (b) In the Southern hemisphere
(a) Northern hemisphere: Rhumb Line is nearer the Equator (South of the Great Circle)
(b) Southern hemisphere: Rhumb Line is nearer the Equator (North of the Great Circle)
In both hemispheres, the Rhumb Line lies nearer to the Equator, while the Great Circle bulges toward the nearer Pole. This is why the GC is the shorter path — it cuts across the "top" of the curved path rather than staying at the lower latitude.
A Rhumb Line — it cuts all other meridians at a constant angle of 0° (it runs exactly N–S, maintaining constant direction).
A semi-Great Circle — it forms one half of a Great Circle (the other half being its anti-meridian), whose plane passes through the Earth's centre.
Distractor Analysis:
(a) Correct but incomplete — a meridian is also a semi-Great Circle.
(b) Correct but incomplete — a meridian is also a Rhumb Line.
Instructor's Note: Source key: c. Verified. No discrepancy.
Q11. A Rhumb Line cuts all meridians at the same angle. This gives:
(b) A line which could never be a Great Circle track
(c) A line of constant direction ✓
▶ Show answer & explanation
Correct Answer: (c) — A line of constant direction
Cutting all meridians at the same angle is equivalent to maintaining a constant compass direction. This is the defining property of a Rhumb Line and the basis of traditional compass navigation.
Distractor Analysis:
(a) Wrong — constant direction gives the longer (Rhumb Line) path, not the shortest (Great Circle).
(b) Wrong — meridians cut all other meridians at 0° (a constant angle) AND are semi-Great Circles. So a Rhumb Line CAN be a Great Circle track.
Instructor's Note: Source key: c. Verified. No discrepancy.
GC always changes direction towards the Equator. Mnemonic: DIID read from North-West clockwise.
Answer Key Summary — No ⚑ Flags
Q
Answer(s)
1
a) 12°40'S b) 109°55'S c) 39°50'N d) 21°55'
2
2 285 NM | 4 232 km
3
a) 49°55'W b) 05°40'W c) 28°50'E d) 56°42'W
4
a) 01°29'N 002°05'W b) 03°36'S 001°13'E c) 00°26'N 005°22'E d) 67°24'N 004°44'E e) 00°01'N 014°36'E
5
a) 90/167 b) 482/893 c) 3 360/6 223 d) 660/1 222 e) 10 800/20 000
6
a) 360°(T) b) No c) Over N Pole, then track reverses to 180°T
7
70°40'N 023°00'W
8
10°00'N 160°09'W
9
a) Nearer Equator (S of GC) b) Nearer Equator (N of GC)
10
c
11
c
DGCA CPL/ATPL General Navigation Study Notes
Chapter 2 — Great Circles, Rhumb Lines & Directions on the Earth Capt. Pankaj Pahil | www.ghostaviator.com For personal study use only. Ghost Aviator Interactive Colour Edition.
