Earth Geometry
Learning Objective
Calculate the distance between places on a great circle.
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DISTANCES BETWEEN TWO PLACES ON THE SAME GREAT CIRCLE
The great circles on the surface of the earth are the equator and all the longitudes.
For all the Great Circles R = 𝟔𝟑𝟕𝟎 km.
The distance between the points AB =
𝛉
𝟑𝟔𝟎
× 𝟐𝛑𝐑.
Where 𝛉 = 𝟔𝟎° + 𝟔𝟎°
= 𝟏𝟐𝟎°
The Circumference of the Earth 𝐂 = 𝟐𝛑𝐑.
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Example
1
Calculate the distance in km between the following pairs of locations:
(a) P (50°S, 100°E) and Q (12°N, 100°E)
(b) A (40°N, 20°W) and B(75°N, 20°W)
(c)
C(0°, 40°W) and D(0°, 20°E)
Take 𝛑 to be 𝟑. 𝟏𝟒𝟐 and R = 𝟔𝟑𝟕𝟎 km
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Solutions
(a) angular distance = 50° + 12°
= 62°
PQ =
=
θ
360
62
360
× 2πR
× 2 × 3.142 × 6370
= 6894 km
(b) angular distance = 75° − 40°
= 35°
θ
AB = 360 × 2πR
35
=
× 2 × 3.142 × 6370
360
= 3892 km
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(c) angular distance = 40° + 20°
= 60°
CD =
=
θ
360
60
360
× 2πR
× 2 × 3.142 × 6370
= 6672 km
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Exercise
1. Beijing, China and Perth have coordinates (40°N, 116°E) and (32°S, 116°E) respectively.
(a) What great circle joins Beijing and Perth?
(b) What is the angular distance between these two cities?
(c) Hence, calculate the shortest distance between Beijing and Perth.
Take 𝛑 to be 𝟑. 𝟏𝟒𝟐 and R = 𝟔𝟑𝟕𝟎 km
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2. The diagram below shows the positions of points A, B, C and D on the surface of the
earth. [Take 𝛑 to be 𝟑. 𝟏𝟒𝟐 and R = 𝟔𝟑𝟕𝟎 km]
48°N
(a) What great circle passes through points C and D?
(b) What is the angular distance between C and D?
A
B
0°
(c) Calculate the shortest distance between C and D.
(d) What great circle passes through points A and D?
D
C
(e) Calculate the great circle distance between A and D.
(f) Calculate the distance between points B and C.
Give your answers correct to the nearest kilometer.
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3. Calculate, to the nearest kilometre, the great circle distances between these cities. [Take 𝛑 to be
𝟑. 𝟏𝟒𝟐 and R = 𝟔𝟑𝟕𝟎 km].
(a) Osaka, Japan (34°N, 135°E) and Alice Springs (23°S, 135°E)
(b) Pretoria, South Africa (25°S, 28°E) and Minsk, Belorussia (54°N, 28°E)
(c) Kampala, Uganda (0°, 32°E) and Macapa, Brazil (0°, 51°W)
(d) Mawson, Antarctica (67°S, 63°E) and Chelyabinsk, Russia (54°N, 63°E)
(e) Port Moresby, Papua New Guinea (9°S, 147°E) and Nyngan (31°S, 147°E)
(f) Geelong (38°S, 144°E) and Ivanhoe (33°S, 144°E)
(g) Leeton (34°S, 146°E) and Townsville (19°S, 146°E)
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Distance on a Small circle
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The Shortest Distance
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