Pin any point on the planet with two numbers, latitude and longitude, then work out how far apart two places are along a great circle. The whole topic runs on one number: 111.2 km per degree.
What even is this? Every spot on Earth has a latitude (how far north or south of the equator) and a longitude (how far east or west of the Greenwich line). The shortest path between two points runs along a great circle, a circle whose centre is the centre of the Earth. This page is the other half of Topic 5, the partner to Time Zones.
If you only do one thing
Distance along a great circle = 111.2 × angular distance. On the same meridian (longitude), the angular distance is the latitude difference. On the same parallel (latitude), multiply by cos θ first, where θ is that latitude.
The 10-minute version
Low energy? Do just this and you have still won the day. Zero is the only fail.
The one move. Two places on the same meridian, one at 20°S and one at 50°S:
Angular distance = 50 − 20 = 30° (same hemisphere, so subtract). D = 111.2 × 30 = 3336 km
Then do Practice Question 1. Screenshot the win for Nat and stop there.
Latitude measures how far north or south you are, from 0° at the equator to 90° at each pole. Lines of equal latitude run east to west and are called parallels. Longitude measures how far east or west you are from the Greenwich meridian (0°), up to 180°. Lines of equal longitude run north to south and are called meridians. Every meridian is half of a great circle.
Parallels (latitude) run east to west; meridians (longitude) run north to south. A and B sit on the same meridian, so the path between them follows that great circle.
A position is written latitude first, then longitude, with a hemisphere letter on each, for example 27°S, 153°E (near Brisbane). The letters matter: they tell you which side of the equator and which side of Greenwich you are on.
2 · Finding the angular distance
Both distance formulas need the angular distance, the angle (in degrees) between the two places measured at the centre of the Earth. How you find it depends on whether the places share a meridian or a parallel, but the hemisphere rule is always the same.
The hemisphere rule
➖
Same side
Both N, both S, both E or both W → subtract the two angles
➕
Opposite sides
One N one S, or one E one W → add the two angles
1
Same meridian (same longitude, different latitude): the angular distance is the difference in latitude. Use the hemisphere rule on the two latitudes.
2
Same parallel (same latitude, different longitude): the angular distance is the difference in longitude. Use the hemisphere rule on the two longitudes.
3 · The two distance formulas
Same meridian (along a line of longitude)
D = 111.2 × angular distance
D = distance in kilometres · angular distance = difference in latitude (degrees)
Same parallel (along a line of latitude)
D = 111.2 × cos θ × angular distance
θ = the latitude of the parallel · angular distance = difference in longitude (degrees)
📘 Straight from the formula book. Both formulas are printed exactly like this: D = 111.2 × angular distance and D = 111.2 cos θ × angular distance. The 111.2 is the number of kilometres in one degree along a great circle. The cos θ only appears for the same-parallel case, because circles of latitude get smaller as you move away from the equator.
⚠️ The classic trap: only use cos θ when the two places are on the same parallel (same latitude). For two places on the same meridian, there is no cos θ, just 111.2 × angular distance. Mixing these up is the most common lost mark.
4 · Worked examples
Worked Example A · same meridian
A yacht sails due south along the 150°E meridian, from a buoy at 20°S to a buoy at 50°S.
Same or opposite? Both latitudes are south, so subtract.
Angular distance = 50 − 20 = 30°
Same meridian, so no cos θ: D = 111.2 × 30
D = 3336 km along the great circle.
Worked Example B · same parallel
Two weather stations both sit on the 30°N parallel, one at 20°E and one at 50°E.
Same or opposite? Both longitudes are east, so subtract.
Angular distance = 50 − 20 = 30° of longitude
Same parallel, so use cos θ with θ = 30°: D = 111.2 × cos 30° × 30
= 111.2 × 0.8660 × 30
D ≈ 2889 km. Notice it is shorter than Example A for the same 30°, because the parallel is smaller than the equator.
Worked Example C · opposite hemispheres
Two islands lie on the same meridian, one at 15°N and one at 25°S.
Same or opposite? One is north, one is south, so add.
Angular distance = 15 + 25 = 40°
Same meridian: D = 111.2 × 40
D = 4448 km.
5 · Practice questions
1. Two towns lie on the same meridian, one at 12°N and one at 48°N. How far apart are they?Tap to reveal
Both north, so subtract: angular distance = 48 − 12 = 36°.
Same meridian, no cos θ: D = 111.2 × 36 = 4003.2 km.
2. Two lighthouses both sit on the 45°N parallel, one at 10°E and one at 30°E. Find the distance between them.Tap to reveal
Both east, so subtract: angular distance = 30 − 10 = 20° of longitude.
Same parallel, use cos θ with θ = 45°: D = 111.2 × cos 45° × 20
= 111.2 × 0.7071 × 20 ≈ 1572.6 km.
3. A ship at 33°S and a port at 12°N lie on the same meridian. How far apart are they?Tap to reveal
One south, one north, so add: angular distance = 33 + 12 = 45°.
Same meridian: D = 111.2 × 45 = 5004 km.
4. Two places on the same meridian are 2780 km apart. What is the angular distance between them?Tap to reveal
5. Two research bases both sit on the 60°S parallel, at 40°E and 100°E. Find the distance, then explain why it is the same as a 30° gap along the equator.Tap to reveal
Both east, subtract: angular distance = 100 − 40 = 60°.
Same parallel, cos θ with θ = 60°: D = 111.2 × cos 60° × 60 = 111.2 × 0.5 × 60 = 3336 km.
Because cos 60° = 0.5, the 60° gap shrinks to an effective 30°, so it matches a 30° gap on the equator (where cos 0° = 1). The further from the equator, the more cos θ shrinks the distance.