← Earth Geometry and Time Zones › Distance on Earth’s Surface
Distance on Earth’s Surface
Key Terms
- Distance along a great circle: d = (πRθ) / 180, where θ = central angle in degrees
- Distance along a parallel (small circle): d = (πrθ) / 180, where r = R cos(φ)
- 1 nautical mile = 1 minute of arc of a great circle = 1.852 km
- For cities on the same meridian: θ = difference in latitudes
- For cities on the equator: θ = difference in longitudes
- For cities at the same latitude (not on equator): use the small circle formula
Great circle arc: d = (πRθ) / 180
Small circle arc (same latitude φ):
d = (πR cos(φ) × Δλ) / 180
where Δλ = longitude difference
Nautical miles:
1 nautical mile = 1 minute of arc = 1.852 km
θ in minutes = θ° × 60
Small circle arc (same latitude φ):
d = (πR cos(φ) × Δλ) / 180
where Δλ = longitude difference
Nautical miles:
1 nautical mile = 1 minute of arc = 1.852 km
θ in minutes = θ° × 60
Worked Example: Perth (32°S, 116°E) and Auckland (37°S, 175°E) are approximately on the same latitude. Find the distance along the parallel of latitude 32°S between them (use longitude difference 175 − 116 = 59°).
r = 6400 × cos(32°) = 6400 × 0.8480 = 5427 km
d = (π × 5427 × 59) / 180 ≈ 5 591 km
Note: The actual great circle distance is shorter (≈ 5 000 km); travel along the parallel is longer than the great circle route.
r = 6400 × cos(32°) = 6400 × 0.8480 = 5427 km
d = (π × 5427 × 59) / 180 ≈ 5 591 km
Note: The actual great circle distance is shorter (≈ 5 000 km); travel along the parallel is longer than the great circle route.
Hot Tip: The great circle route is always shorter than (or equal to) the small circle route between two points. When a question asks for the “shortest distance”, use the great circle. When it says “along the parallel”, use the small circle formula.
Three Common Cases
Most distance problems fall into one of three categories:
- Same meridian (same longitude): travel along a great circle. Use θ = latitude difference.
- Same parallel (same latitude): travel along a small circle. Use r = R cos(φ) and θ = longitude difference.
- Equator (latitude = 0°): the equator is a great circle. Use r = R and θ = longitude difference.
Nautical Miles
A nautical mile is defined as 1 minute of arc (1/60 of a degree) along a great circle. Since 1 degree = 60 nautical miles:
distance (nm) = θ° × 60
And 1 nm = 1.852 km.
Distinguishing Great Circle vs Small Circle Routes
Airlines use great circle routes because they are the shortest path. Sailors historically used parallel (rhumb line) routes because they are easier to navigate (constant compass bearing), even though they are slightly longer.
Strategy: Always identify which type of path (great circle or small circle) the question is asking about before applying the formula. Read the question carefully for clues like “shortest route” vs “along the parallel”.
Mastery Practice
- Fluency Two cities are on the same meridian at latitudes 10°N and 50°N. Find the great circle distance between them.
- Fluency Two cities lie on the equator at 45°W and 75°E. Find the distance between them along the equator.
- Fluency Convert a great circle arc of 30° into: (a) kilometres (b) nautical miles.
- Fluency Two cities are both at 50°N, separated by 40° of longitude. Find the distance along the parallel of latitude.
- Understanding Cairns, Australia is at approximately (17°S, 146°E) and Port Moresby, Papua New Guinea is at approximately (9°S, 147°E). Treating them as on the same meridian, find the great circle distance.
- Understanding Two cities are both at 60°S, separated by 90° of longitude. Find: (a) the distance along the parallel (b) the great circle distance along the meridian if they were instead at 60°S, 0° and 60°S, 90° (note: these require different approaches).
- Understanding Express a distance of 3 000 km as an angle (in degrees) along a great circle.
- Understanding A ship travels due east along latitude 30°S from 20°E to 80°E. How far does it travel? Is this a great circle route?
- Problem Solving London is at (51.5°N, 0°) and Moscow is at (55.8°N, 37.6°E). Treating them as on the same meridian (use the longitude of Moscow as an approximation), find the great circle distance from London to Moscow.
- Problem Solving A pilot flies from (40°N, 30°E) to (40°N, 90°E) by two routes:
- (a) Along the parallel of latitude (small circle route)
- (b) Along the meridian to the equator, then along the equator, then up the other meridian (great circle segments). Calculate each distance and determine which is shorter.