# What is the shortest distance between two points (61^@40.177',33^@23.101') and (59^@26.266',13^@03.807') ("latitude","longitude") on earth?

Mar 31, 2017

Great circle distance is $1411.55$ miles.

#### Explanation:

First of all here angles (latitudes and longitudes) are given here in degrees and minutes and hence need to converted into degrees (with decimals) so that I can use scientific calculator provided with MS Windows. Further, I will be using up to six places (or more) of decimal for accuracy.

Hence ${33}^{\circ} 23.101 ' = {\left(33 + \frac{23.101}{60}\right)}^{\circ} = {33.385016}^{\circ}$.

Similarly ${13}^{\circ} 03.807 ' = {13.06345}^{\circ}$, ${61}^{\circ} 40.177 ' = 61.669616$ and ${59}^{\circ} 26.266 ' = {59.437767}^{\circ}$

Further, although we are using degrees as longitudes and latitudes are available in degrees, $d$ should be found in radians, to get great circle distance (GCD - it is the shortest distance between two points on the surface of a sphere, here earth) and then this distance would be $d \times R$. Now using the formula,

d=cos^(-1)(sin33.385016^@sin13.06345^@+cos33.385016^@cos13.06345^@ cos(61.669616 – 59.437767)^@

= ${\cos}^{- 1} \left(0.5502624 \times 0.2260299 + 0.8349918 \times 0.97412 \times \cos \left(2.231849\right)\right)$

= ${\cos}^{- 1} \left(0.5502624 \times 0.2260299 + 0.8349918 \times 0.97412 \times 0.9992414\right)$

= ${\cos}^{- 1} \left(0.12437576 + 0.8127652\right)$

= ${\cos}^{- 1} \left(0.93714096\right)$

= $0.35645154$ - in radians

$G C D = 0.35645154 \times 3960 = 1411.55$ miles.