MATH 117 Arclength Around the Earth Part 2 φ R R r Equator

Dr. Neal, WKU
MATH 117
Arclength Around the Earth
Part 2
The radius of the Earth is about R = 3963.2 miles. Therefore, the circle defined by the
equator also has this radius R. But if we go North or South to a particular latitude ! ,
then we can define a concentric circle, parallel to the equator, that has a smaller radius r .
How do we find r ?
North
Pole
r
R
R
!
Equator
R ! 3963.2 miles
We simply set up a right triangle to solve for r .
r
r
R
R
!
!
r
R
cos ! =
r
R
and
r = Rcos !
On Earth, r ! 3963.2cos "
So now we can find the direct East/West distance between two points at the same
latitude using s = ! r = ! " 3963.2cos # , where ! is the radian angle between the points
and ! is the common latitude.
Dr. Neal, WKU
Example. Find the direct East/West distances between the following pairs of points:
(a) 38º 15! N, 42º 36! W
and
(b) 44º 18! S, 32º 12 ! E
and
38º 15! N, 22º 30 ! E
44º 18! S, 15º 54! E
Solution. (a) The radius of the East/West circle at 38º 15! N latitude is given by
#
15 &
r ! 3963.2cos(38º 15") = 3963.2cos% 38 + ( = 3963.2 cos(38.25) ! 3112.368 miles
$
60 '
Next, the angle between 42º 36! W and 22º 30 ! E is given by the sum:
"
36 % "
30 %
42º 36! + 22º 30! = $ 42 + ' + $ 22 + ' = 65.1º
#
60 & #
60 &
So the direct East/West distance between the points is given by
65.1º !
"
! 3963.2 cos (38. 25) ≈ 3536.3 miles
180º
(b) For a southern latitude, we take the angle to be negative (4th Quadrant), but the
cosine will still be positive value. So the radius at 44º 18! S = 44.3º S = –44.3º is given by
r ! 3963.2cos("44.3) ! 2836.43344 miles
The angle between 32º 12 ! E and 15º 54! E is given by the difference:
#
12 & #
54 &
32º 12 ! " 15º 54 ! = % 32 + ( " % 15 + ( = 16.3º
$
60 ' $
60 '
So the direct East/West distance is given by
16. 3º !
"
! 3963.2 cos( #44.3 ) ≈ 806.933 miles
180º