FMSP
Year 10 Team Mathematics Competition
2012
Introduction to Distances
on the Earth
Introduction to Distances on the Earth
The main ideas are:
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Great circles
Small circles
Position on the Earth
(Latitude/Longitude)
Nautical miles
Shortest distance between
two points on a great circle
Distance between two
points on a small circle
Before you continue, download the template and
instructions to make a model of the Earth.
Introduction to Distances on the Earth
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The shape of the
Earth closely resembles a
flattened sphere
(a spheroid) with an
equatorial radius of
6,378 km, whilst the
distance from the centre of
the spheroid to each pole is
6357 km.
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We are going to model the
earth as a perfect sphere.
Introduction to Distances on the Earth
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So the equator and lines of
longitude all lie on great
circles. These are circles
with a radius equal to the
radius of the Earth (approx.
6400 km). This also means
that the plane of the circle
intersects with the centre of
the Earth.
Introduction to Distances on the Earth
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A nautical mile (NM) is
defined as the distance
travelled on a great circle
when moving through 1
minute of arc.
There are 360o in a circle
and 60 minutes in a degree.
So there are 360 x 60 =
21600' in a circle.
So the distance around the
equator is 21600 NM.
Introduction to Distances on the Earth
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So to calculate the shortest
distance from 60o W to 10o E
along the equator (a great
circle):
Smallest angle along the
equator between the two
positions is 60 + 10 =70o
Shortest distance
= 70 x 60
= 4200 NM
Remember to
convert degrees
into minutes.
Introduction to Distances on the Earth
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Calculate the shortest distance
between (0o N, 42o 15' W) and
(0o N, 127o 32’ W)
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Smallest angle along the
equator between the two
positions is 85o 17'.
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Shortest distance
= 85 x 60 + 17
= 5117 NM
Remember to
convert degrees
into minutes.
Introduction to Distances on the Earth
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All circles which pass through
the North and South poles are
great circles.
Lines of longitude lie on great
circles.
Directly opposite meridians
form great circles.
So the Greenwich (Prime)
meridian 0o W and the
International Date Line 180o W
form a great circle.
Introduction to Distances on the Earth
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The longitudes 90o W and 90o E
form a great circle as they are
opposite each other.
Calculate the shortest distance
between (70o N, 90o W) and
(50o N, 90o E):
Smallest angle along the great
circle is 20 + 40 = 60o.
Shortest distance
= 50 x 60
= 3000 NM.
Smallest angle
passes over the
North Pole
Introduction to Distances on the Earth
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Small circles are parallel to the
equator.
Lines of latitude are small
circles.
The latitude of the equator is
0o.
The latitude of the North pole
is 90o N.
The latitude of the South Pole
is 90o S.
All other latitudes lie between
these two angles.
Introduction to Distances on the Earth
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As you travel around a line of
latitude (small circle) the
distance travelled is shorter
than the distance covered on
the Equator (great circle) for a
given angle.
Radius of the small circle (r)
Radius of the Equator (R)
Latitude angle 
Then r = R cos 
So the small circle smaller than
the great circle by a factor of
cos .
N
r
Latitude 41o N
41o
Equator
R
S
r
N
R
Introduction to Distances on the Earth
N
r
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Calculate the distance along the line
of latitude from (41o N, 36o W) to
(41o N, 155o E) .
Smallest angle is not 191o, but 169o.
Converting this angle into minutes:
Angle = 169 x 60
= 10140'.
Since the measurement is along a
small circle the distance is reduced
by a factor of cos 41o.
Distance in nautical miles
= 10140 x cos 41o
= 7653 NM
Latitude 41o N
41o
Equator
R
S
155o E
N
36o W
0o
This is not the shortest distance!
Remember: The shortest distance is on a
great circle passing through these two
points, centred at the Earth’s centre.
Introduction to Distances on the Earth
N
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From (20o S, 15o E) I travel West
3000 NM. What is my new position?
Equator
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Since we are moving along a small
circle, the angle will be larger than
3000'.
Angle = 3000 ÷ cos 20o
= 3193'
= 53o 13'
Moving West from our original
position will take us past the
Greenwich meridian to a westerly
point at an angle of (53o 13' - 15o).
New position is (20o S, 38o 13' W)
Latitude 20o S
S
Use the scale factor to convert
from the small circle to the
great circle, as if we are
moving along the Equator.
? oW
0o
15o E
Introduction to Distances on the Earth
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For further reading and questions, you can copy and
paste this link:
http://books.google.co.uk/books?id=vk0Nt1cN_9MC&pg=PA146&dq=Earth+as+a+Sphere+mathematics&hl=en&sa=X&ei=k8rrTu_0B
cOSiQfC6eWzBw&sqi=2&ved=0CDwQ6AEwAQ#v=onepage&q=Earth%20as%20a%20Sphere%20mathematics&f=false