The Great Circle Distance

5 min readJun 11, 2023

Finally. Got the time to write a post after a looooong time. (Okay I will admit it. It is not that I didn’t have time. I was just lazy to type)

In this post, I will covering a topic known as the Great Circle Distance which is related to my previous post on latitudes & longitudes to some extent

What Is It?

The great circle is the name given to the circle that contains the shortest point between any 2 points in a sphere

Simply put, it is the shortest distance between any 2 points on the Earth

On a simple piece of paper, consider any 2 points. The shortest distance between them is a straight line connecting both the points.

But imagine a sphere like the Earth. How do you calculate the shortest distance between any 2 points. Consider any 2 points on a sphere. Imagine a piece of string connecting both the cities. You might think it must be a straight line like the one in left.

But that is not the case. Try unwrapping the sphere and you will get something similar to the one in the right

What you thought was a straight line was not actually straight. It was actually bent. Why is this so. This is because it is next to impossible to represent shapes like spheres accurately in 2 dimensions

How To Calculate It?

The formula for calculating the Great Circle Distance defers for a sphere and an ellipse. The one for an ellipse is quite complex and formula for a sphere is comparatively easier. In this post, we will be dealing with only a sphere, which is closest in shape to the earth

Consider 2 points A & B whose co-ordinates are

A — (Xa, Ya)

B — (Xb, Yb)

Where Xa & Xb are the latitudes

Ya & Yb are the longitudes

The Great Circle Distance is given by

where “r” is the radius of the sphere

The most important point to note is that all angles are in radians

An Example

Consider 2 cities Chennai & Mumbai

Mumbai’s co-ordinates are 19.08 N & 72.88 E

Chennai’s co-ordinates are 13.08 N & 80.27 E

The convention followed is:

If latitude is north, it is +ve, if it is south, it is -ve

If longitude is east, it is +ve, if it is west, it is -ve

For both Chennai & Mumbai, since both latitudes are north & both longitudes are east, they remain positive

To convert degrees to radians, we multiply it by pi=3.14 & divide by 180, which gives us

By the above formula

Xa = 0.333 & Xb = 1.272

Ya = 0.228 & Yb = 1.401

r = radius of earth = 6371 km

Let us tackle the formula piece by piece

Starting from the first term within the bracket:

sin(Xa)*sin(Xb)=sin(0.333)*sin(1.272) = 0.07397

Moving on to the 2nd term

cos(Xa)*cos(Xb)*cos(Ya-Yb)=cos(0.333)*cos(1.272)*cos(0.228–1.401) = 0.9129

Adding both and taking inverse cosine gives

Multiplying this with the radius of the earth(r) = 6371 km gives

d = 6371*0.1622 = 1033 km

Hence, we find that the great circle distance between Mumbai & Chennai is 1033 km

Across The World

Let us consider 2 points on opposite ends of the world like New York in USA & Sydney in Australia

New York: 40.71 N & -74 W

Sydney: 33.87 S & 151.21 E

Performing a similar calculation gives the great circle distance as 15,989 km

(If you have read my previous post on time differences, you could try calculating the time difference between the 2 cities)

Conclusion — Is It Really “Great”?

Consider 2 cities Dubai & New York

The below graph shows both the Great Circle Distance (in red) & the Rhumbline (in green). In simple terms, the Rhumbline is the distance you get when you connect both the origin and destination in a straight line.

(Source: https://www.kavas.com/blog/great-circle-and-rhumbline.html. You can go to this website and try out the great circle distance for different pairs of cities)

We see that the Great Circle Distance is the shorter distance though it might seem otherwise. This is the route that most aircrafts use while flying between 2 cities. This helps aircraft companies save on both fuel and time. In that sense we can say that the Great Circle Distance is “great”

Refer below to the actual route taken by Emirates flight EK203 from Dubai to New York which closely follows the Great Circle route. (Source: flightradar24 website)