Great Circle Distance Formula

A Great Circle is the largest possible circle that can be drawn on the surface of a sphere. It is formed by the intersection of the sphere and a plane that passes through the center of the sphere, dividing the sphere into two equal hemispheres.

The figure below depicts a sphere and circles of different radii and sizes that can be drawn on its surface. 

Shortest Distance:

• A great circle represents the shortest path between two points on a sphere, often used in aviation and navigation (e.g., flying from one city to another).
• They are important in navigation, as the path along a great circle is the most efficient route between distant locations. Sometimes, great circles are also known as Romanian Circles.

In this article, we will explore the applications of the great circle, derive its formula, and work through solved examples.

Application of the Great Circle

The application of the great circle formula is utilized in the navigation of aircraft or ships, which is one of the few instances where it is applied. Because the Earth is spherical, the great circle formula aids navigation by determining the shortest distance within the sphere. 

As can be seen below, the great circle is the one that shares the centre with the main sphere itself. The diameter of both the sphere and the great circle coincides.

Great Circle Formula

The great circle formula is given as follows:

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

Where, 

• r depicts the earth's radius, 
• a and b depict the latitude 
• while the longitudes are depicted by x and y.

Sample Problems

Question 1. Given that the radius of a sphere is 4.7 km, latitude being (45°, 32°) and longitude (24°,17°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 4.7 km or 4700 m, a, b= 45°, 32° and x, y = 24°,17°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians = Degrees × π​/180
a = 45° = 45 × π/180 = 0.7854
b = 32° = 32 × π/180 = 0.5585
x = 24° = 24 × π/180 = 0.4189
y = 17° = 17 × π/180 = 0.2967
x - y = 0.4189 - 0.2967 = 0.1222

Now calculate all trigonometric values:

• cosa = cos(0.7854) = 0.7071
• cos⁡b = cos⁡(0.5585) = 0.8480
• cos⁡(x−y) = cos⁡(0.1222) = 0.9925
• sin⁡a = sin⁡(0.7854) = 0.7071
• sin⁡b = sin⁡(0.5585) = 0.5314

Now, d = 4700⋅cos⁻¹(0.7071⋅0.8480⋅0.9925 + 0.7071⋅0.5314)
= 4700⋅cos⁻¹(0.9731)
= 4700 × 0.2326 = 1093.2

⇒ d = 1093.2 m

Question 2. Given that the radius of a sphere is 5 km, latitude being (25°, 34°) and longitude (48°,67°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 5 km or 5000 m, a, b= 25°, 34° and x, y = 48°,67°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians = Degrees × π​/180
a = 25° = 25 × π/180 = 0.4363
b = 34° = 34 × π/180 = 0.5934
x = 48° = 48 × π/180 = 0.8378
y = 67° = 67 × π/180 = 1.1694
x - y = 1.1694 - 0.8378 = 0.3316

Now calculate all trigonometric values:

• cosa = cos(0.4363) = 0.9063
• cos⁡b = cos(0.5934) = 0.8290
• cos⁡(x−y) = cos(0.3316) = 0.9449
• sin⁡a = sin(0.4363) = 0.4226
• sin⁡b = sin(0.5934) = 0.5299

Now, d = 5⋅cos⁻¹(0.9063)⋅(0.8290)⋅(0.9449)+(0.4226)⋅(0.5299)
d = 5⋅cos⁻¹(0.7185+0.2248) =5⋅cos⁻¹(0.9433)
= 5⋅0.3316 = 1.658 km

⇒ d = 1.658 km

Question 3. Given that the radius of a sphere is 10 km, latitude being (55°, 86°) and longitude (28°,70°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 10 km , a, b= 55°, 86° and x, y = 28°,70°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians = Degrees × π​/180
a = 55° = 55 × π/180 = 0.9599
b = 86° = 86 × π/180 = 1.5010
x = 28° = 28 × π/180 = 0.4887
y = 70° = 70 × π/180 = 1.2210
x - y = 1.2210−0.4887=0.7323

Now calculate all trigonometric values:

• cosa = cos(0.9599) = 0.5775
• cos⁡b = cos(1.5010) = 0.0707
• cos⁡(x−y) = cos(0.7323) = 0.7451
• sin⁡a = sin(0.9599) = 0.8169
• sin⁡b = sin(1.5010) = 0.9975

Now, d = 10⋅cos⁻¹(0.5775)⋅(0.0707)⋅(0.7451)+(0.8169)⋅(0.9975)
d = 10⋅cos⁻¹(0.0310 + 0.8150) =10⋅cos⁻¹(0.8460)
=10⋅0.5567 = 5.567 km

⇒ d = 5.567 km

Question 4. Given that the radius of a sphere is 7 km, latitude being (55°, 86°) and longitude (28°,70°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 7 km , a, b= 55°, 86° and x, y = 28°,70°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians=Degrees × π​/180
a = 55° = 55 × π/180 = 0.9599
b = 86° = 86 × π/180 = 1.5010
x = 28° = 28 × π/180 = 0.4887
y = 70° = 70 × π/180 = 1.2210
x - y = 1.2210−0.4887=0.7323

Now calculate all trigonometric values:

• cosa = cos(0.9599) = 0.5775
• cos⁡b = cos(1.5010) = 0.0707
• cos⁡(x−y) = cos(0.7323) = 0.7451
• sin⁡a = sin(0.9599) = 0.8169
• sin⁡b = sin(1.5010) = 0.9975

Now, d = 7⋅cos⁻¹(0.5775)⋅(0.0707)⋅(0.7451)+(0.8169)⋅(0.9975)
d = 7⋅cos⁻¹(0.0310 + 0.8150) =7⋅cos⁻¹(0.8460)
=7⋅0.5567 = 3.9627 km

⇒ d = 3.9627 km

Question 5. Given that the radius of a sphere is 4 km, latitude being (25°, 34°) and longitude (48°,67°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 4 km , a, b= 25°, 34° and x, y = 48°,67°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians=Degrees × π​/180
a = 25° = 25 × π/180 = 0.4363
b = 34° = 34 × π/180 = 0.5934
x = 48° = 48 × π/180 = 0.8378
y = 67° = 67 × π/180 = 1.1694
x - y = 1.1694 − 0.8378 = 0.3316

Now calculate all trigonometric values:

• cosa = cos(0.4363) = 0.9063
• cos⁡b = cos(0.5934) = 0.8290
• cos⁡(x−y) = cos(0.3316) = 0.9451
• sin⁡a = sin(0.4363) = 0.4226
• sin⁡b = sin(10.5934) = 0.5592

Now, d = 4⋅cos⁻¹[0.9063⋅0.8290⋅0.9451+0.4226⋅0.5592]
d = 4⋅cos⁻¹(0.7085+0.2363) = 4⋅cos⁻¹(0.9448)
=4⋅0.3300 = 1.32 km

⇒ d = 1.32 km

Question 6. Given that the radius of a sphere is 16 km, latitude being (45°, 32°) and longitude (24°,17°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 16 km , a, b = 45°, 32° and x, y = 24°,17°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians=Degrees × π​/180
a = 45° = 45 × π/180 = 0.7854
b = 32° = 32 × π/180 = 10.5585
x = 24° = 24 × π/180 = 0.4189
y = 17° = 17 × π/180 = 0.2967
x - y = 0.1222

Now calculate all trigonometric values:

• cosa = cos(0.7854) = 0.7071
• cos⁡b = cos(0.5585) = 0.8480
• cos⁡(x−y) = cos(0.1222) = 0.9925
• sin⁡a = sin(0.7854) = 0.7071
• sin⁡b = sin(0.5585) = 0.5299

Now, d = 16⋅cos⁻¹[0.7071⋅0.8480⋅0.9925 + 0.7071⋅0.5299]
d = 16⋅cos⁻¹[0.5970 + 0.3746] = 16⋅cos⁻¹(0.9716)
=16⋅0.2390 = 3.824km

⇒ d = 3.824 km

Question 7. Given that the radius of a sphere is 18 km, latitude being (45°, 32°) and longitude (24°,17°), find the distance of the great circle.

Solution:

The great circle formula is given by: d = rcos^-1[cos a cos b cos(x-y) + sin a sin b].

Given: r = 18 km , a, b= 45°, 32° and x, y = 24°,17°.
Substituting the values in the above formula, we have:

Convert degrees to radians
Radians=Degrees × π​/180
a = 45° = 45 × π/180 = 0.7854
b = 32° = 32 × π/180 = 0.5585
x = 24° = 24 × π/180 = 0.4189
y = 17° = 17× π/180 = 0.2967
x - y = 0.1222

Now calculate all trigonometric values:

• cosa = cos(0.7854) = 0.7071
• cos⁡b = cos(0.5585) = 0.8480
• cos⁡(x−y) = cos(0.1222) = 0.9925
• sin⁡a = sin(0.7854) = 0.7071
• sin⁡b = sin(0.5585) = 0.5299

Now, d = 18⋅cos⁻¹[0.7071⋅0.8480⋅0.9925 + 0.7071⋅0.5299]
d = 18⋅cos⁻¹[0.5970+0.3746] = 18⋅cos⁻¹(0.9716)
=18⋅0.2390 = 4.302 km

⇒ d = 4.302 km