Distance along a small circle

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small circle

All latitudes except the equator are small circles. The path joining two points on the same latitude forms an arc of the circle of that latitude.

The radius r of a latitude is given by r=R cos θ where θ is the angle of latitude.

Along the small circle defined by θ^oN or θ^oS, an arc of length 60 cos θ nm subtends an angle of 1^o at the center of that latitude. That is;

1^o = 60 cos θ nm on a latitude θ^oN or θ^oS. Also note that 1 nm = 1.853 km.

Consider two positions on the surface of the earth illustrated below.

The distance AB will be given by AB = (x/360)^o x 2𝝅r= (x/360)^o x 2𝝅R Cos θ .

that is; r = Rcosθ

θ is the angle of latitude and x is the value of angle difference between the longitude of the two points A and B.

Find the distance between the following pair of points in km given that 𝝅 = 22/7 and R = 6370 km.

(a) R(70^oS, 35^oW), S(70^oS, 80^oE)

(b) R(14^oN, 100^oE), S(14^oN, 10^oE)