Great Circle Formula

The great circle is the largest circle that can be drawn on the surface of a sphere. A great circle is a section of a sphere that contains the diameter of the sphere and the diameter is the shortest distance between any two points on the sphere surface is the great circle distance. The great circle is also known as Romanian Circle. One of the few examples where the application of the great circle formula is used is in the navigation of aircraft or ships. As we know the Earth is somewhat spherical, the great circle formula helps in navigating as we come to know for the shortest distance in the sphere.

What is the Great Circle Formula?

Great circle formula can be expressed as,

\(d=r \cos^{-1}[\cos a \cos b \cos(x - y) + \sin a \sin b]\)

where,

• r = Radius of the earth
• a, b=  Latitude
• x, y = Longitude 

Note: Sphere's diameter coincides with the diameter of the great circle.

Let's take a quick look at a couple of examples to understand the great circle formula, better.

Examples Using Great Circle Formula

Example 1: What will be the length of the great circle if the radius of the sphere is 5 km, the latitude is (25^o, 34^o) and the longitude is (48^o, 67^o).

Solution:      

To find: Great circle distance.

Given: 
r = 5 km 
a = 25^o = 0.418 rad, and  b = 34^o= 0.59 rad
x = 48^o = 0.83 rad, y = 67^o = 1.16 rad

Using great circle formula, 
\( \begin{align} d &= r \cos^{-1}[\cos a \cos b \cos(x - y) + \sin a \sin b]  \\ d &= 5 \times \cos^{-1} [ 0.913 \times 0.83 \times 0.945 +  0.406 \times 0.556] \\ &= 5 \times 2.793 \end{align} \)

Answer: Hence, the great circle length is 94.928 km.

Example 2: What will be the length of the great circle if the radius of the sphere is 10 km, the latitude is (55^o, 86^o) and the longitude is (28^o, 70^o).

Solution:      

To find: Great circle distance.

Given: 
r = 10 km
a = 55^o,= 0.959 rad,  b = 86^o= 1.5

x = 28^o=  0.48 y = 70^o = 1.22

Using Great Circle Formula, 
\[ \begin{align} d &= r \cos^{-1}[\cos a \cos b \cos(x - y) + \sin a \sin b]  \\ d &= 10 \times \cos^{-1} [ 0.573 \times 0.0707 \times 0.743 +  0.819 \times 0.997 ]  \\ &= 10 \times 0.562\end{align} \]

Answer: Hence, the great circle distance is 5.62 km.