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Dr. Neal, WKU
MATH 117
Arclength Around the Earth
Part 1
The radius of the Earth is about 3963.2 miles. Longitude measures the East/West angle
! from the Prime Meridian, which runs vertically from the North Pole to the South Pole
and passes just east of London, England. Latitude measures the North/South angle !
measured from the Equator.
L = London
70º N
L
West
"
!
0º
50º E
Equator
Prime
Meridian
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Case 1: If we know the longitudes of two points on the Equator, then we can find the
distance between them using
d =! "
#
" 3963.2 miles
180º
where ! is the (degree) angle between the points.
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Example 1. Find the distance between the following pairs of points on the Equator:
(a) 40º 42! W to 52º 24 ! E
(b) 40º 42! E to 52º 24 ! E
Dr. Neal, WKU
Solution. (a) Because one longitude is West and the other is East, we add these longitude
angles to find the total angle in between:
! = 40º 42! + 52º 24 ! = 92º 66 ! = 93º 6! = 93 +
6
= 93.1º
60
So the distance between these points is
d = 93.1º !
"
! 3963.2 ≈ 6439.8 miles
180º
(b) Now both longitude angles are on the same side of Prime Meridian; so we subtract
them to find the angle in between:
! = 52º 24 ! " 40º 42 ! = 51º 84 ! " 40º 42! = 11º 42! = 11.7º
So the distance between these points is
"
! 3963.2 ≈ 809.3 miles
180º
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d = 11.7º !
Note: When ! = 360º, then we obtain the entire circumference of the Earth:
"
! 3963.2 = 2" ! 3963.2 ≈ 24,901.52 miles
180º
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C = 360º !
Case 2: If two points are on the same meridian (i.e., directly North/South of each other
with the same longitude angle), then we can find the distance between them by using
the angle between the latitudes.
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Example 2. Find the distance between the following pair of points:
50º E, 30º 56 ! N to 50ºE, 44º 20! N
Solution. Both points are at 50º E, so they are directly North/South of each other.
Because they are both North, we subtract the latitudes to find the angle in between:
! = 44º 20" # 30º 56" = 43º 80 " # 30º 56 " = 13º 24" = 13 +
So the distance between these points is
d = 13. 4º !
"
! 3963.2 ≈ 926.89 miles
180º
24
= 13.4º
60