Latitude and Longitude: Distances on the Earth's Surface

By Engr. Henry Akpan

Locating Places on a Round Earth

Every place on Earth can be pinpointed by two numbers, its latitude and longitude. Pilots, ship captains, GPS receivers and weather forecasters all use this grid. Once you know where two places are, you can calculate the distance between them along the curved surface of the Earth, and even the difference in their local times. This tutorial covers latitude and longitude (the earth as a sphere) as tested in WAEC and NECO mathematics: great and small circles, latitude and longitude, distances along a meridian and along a parallel of latitude, the radius of a parallel, nautical miles, time differences, and journeys over the poles, with fully worked examples.

The Earth as a Sphere

For these calculations, the Earth is treated as a perfect sphere of radius R = 6400 km. WAEC questions often say to take π = 22/7 (or 3.142), and you should always use the value the question gives.

  • A great circle is any circle on the sphere whose centre is the centre of the Earth, such as the equator or any circle through both poles. Its radius equals the Earth's radius R. The shortest route between two points on a sphere is always along a great circle.
  • A small circle is any circle on the sphere whose centre is not the Earth's centre. Every parallel of latitude except the equator is a small circle.

Latitude

The latitude of a place is the angle, measured at the centre of the Earth, between the place and the plane of the equator, north (N) or south (S) of it.

  • The equator has latitude 0°. The North Pole is 90°N, and the South Pole is 90°S.
  • All the places with the same latitude lie on a parallel of latitude, a circle parallel to the equator. Parallels get smaller towards the poles.

Longitude

A meridian is a semicircle from the North Pole to the South Pole. The Greenwich meridian (the prime meridian), through Greenwich in London, has longitude 0°. The longitude of a place is the angle, measured at the centre of the Earth in the plane of the equator, between the Greenwich meridian and the meridian through the place, east (E) or west (W) of Greenwich, up to 180°.

A place is written with its latitude first, for example (40°N, 30°E).

Finding the Difference in Latitude or Longitude

  • If two places are on the same side of the equator (both N or both S), subtract their latitudes. If they are on opposite sides, add them.
  • Longitude works the same way: subtract if both are E or both are W, and add if one is E and the other W. If the sum is more than 180°, subtract it from 360° to get the shorter way round.

For example, the latitudes 10°S and 35°N differ by 10 + 35 = 45°, while the longitudes 20°E and 50°E differ by 30°.

Distance Along a Meridian (Great Circle)

Two places on the same meridian (with the same longitude) lie on a great circle of radius R. The distance between them is an arc of that circle:

Distance = (θ ÷ 360°) × 2πR, where θ is the difference in latitude.

Worked example 1

Find the distance along the meridian between P(10°S, 20°E) and Q(35°N, 20°E). (Take π = 22/7 and R = 6400 km.)

Difference in latitude = 10 + 35 = 45°.

Distance = (45/360) × 2 × (22/7) × 6400 ≈ 5029 km.

A plane flying this route at 800 km/h would take about 5029 ÷ 800 ≈ 6.3 hours, or roughly 6 hours 17 minutes.

Worked example 2: finding a new latitude

A ship at 20°N sails due south for 1500 km. Find its new latitude. (Take π = 22/7.)

The angle moved through is θ = 1500 × 360 ÷ (2 × (22/7) × 6400) ≈ 13.4°. Its new latitude is 20° − 13.4° = 6.6°N.

The Radius of a Parallel of Latitude

A parallel of latitude α° is a small circle whose radius r is smaller than R. Looking at a cross-section of the Earth through the poles, the radius of the parallel, the Earth's radius and the angle of latitude form a right-angled triangle, and:

r = R cos α

  • At the equator, α = 0° and cos 0° = 1, so r = R.
  • At 60°, cos 60° = ½, so r = 3200 km.
  • At the poles, α = 90° and r = 0.

Distance Along a Parallel of Latitude

Two places on the same parallel of latitude α are separated by an arc of the small circle of radius R cos α:

Distance = (θ ÷ 360°) × 2πR cos α, where θ is the difference in longitude.

Worked example 3

Two towns lie on latitude 60°N, at longitudes 20°E and 40°W. Find the distance between them along the parallel. (Take π = 22/7.)

Difference in longitude = 20 + 40 = 60°. The radius of the parallel is 6400 × cos 60° = 3200 km.

Distance = (60/360) × 2 × (22/7) × 3200 ≈ 3352 km.

Worked example 4

Find the distance along the parallel between P(40°N, 30°E) and Q(40°N, 50°W). (Take π = 22/7.)

Difference in longitude = 30 + 50 = 80°. r = 6400 cos 40° ≈ 4903 km.

Distance = (80/360) × 2 × (22/7) × 4903 ≈ 6848 km.

The same 80° difference along the equator would be (80/360) × 2 × (22/7) × 6400 ≈ 8939 km. The distance for a given difference in longitude shrinks as you move towards the poles.

Nautical Miles

Navigators measure distances in nautical miles (nm). One nautical mile is the length of an arc of a great circle that subtends one minute of angle (1/60 of a degree) at the centre of the Earth. One nautical mile is about 1.85 km, and a speed of one nautical mile per hour is called a knot.

  • Along a meridian: distance = 60 × θ nautical miles, where θ is the difference in latitude in degrees.
  • Along a parallel of latitude α: distance = 60 × θ × cos α nautical miles, where θ is the difference in longitude.

Worked example 5

Express the distances in examples 1 and 4 in nautical miles.

Example 1: 60 × 45 = 2700 nm. Example 4: 60 × 80 × cos 40° ≈ 3677 nm.

Nautical miles make calculations very simple, because they need no value of π or of R.

Longitude and Time

The Earth turns through 360° in 24 hours, which is 15° per hour, or 1° every 4 minutes. Places further east see the Sun rise earlier, so their local time is ahead.

Time difference = difference in longitude × 4 minutes

Worked example 6

When it is 12 noon at P(40°N, 30°E), what is the local time at Q(40°N, 50°W)?

The difference in longitude is 80°, so the time difference is 80 × 4 = 320 minutes = 5 hours 20 minutes. Q is west of P, so its time is behind: 12:00 − 5 h 20 min = 6:40 a.m.

Official time zones round these differences to whole hours (or sometimes half-hours) for convenience. Nigeria, for example, uses West Africa Time, one hour ahead of Greenwich Mean Time.

Journeys Over the Pole

When two places on the same parallel have longitudes that differ by exactly 180°, they lie on the same great circle through the pole. The route over the pole is then shorter than the route along the parallel.

Worked example 7

X is at (70°N, 10°E) and Y is at (70°N, 170°W). Compare the distance along the parallel with the distance over the North Pole. (Take π = 22/7.)

Difference in longitude = 10 + 170 = 180°, so the places are on opposite meridians.

Along the parallel: r = 6400 cos 70° ≈ 2189 km, so the distance is half the parallel's circumference: (22/7) × 2189 ≈ 6879 km.

Over the pole: each place is 90° − 70° = 20° from the pole, so the angle at the centre along the great circle is 20° + 20° = 40°. The distance is (40/360) × 2 × (22/7) × 6400 ≈ 4470 km.

The polar route is about 2400 km shorter. That is why many long-haul flights between Europe, North America and Asia fly over the Arctic.

Common Mistakes with Latitude and Longitude

  • Using R instead of R cos α for distances along a parallel of latitude.
  • Subtracting when the places are on opposite sides of the equator or of Greenwich. Add instead.
  • Using the difference in longitude for a distance along a meridian. Along a meridian, use the difference in latitude.
  • Getting the time difference the wrong way round. East is ahead, and west is behind.
  • Ignoring the instruction about π, such as using 3.142 when the question says 22/7.

Latitude and Longitude: Practice Questions

Take π = 22/7 and R = 6400 km.

  1. Find the radius of the parallel of latitude 45°N.
  2. Find the distance along the meridian between A(15°N, 8°E) and B(10°S, 8°E).
  3. Find the distance along the parallel between C(45°N, 10°W) and D(45°N, 20°E).
  4. Find the distance in nautical miles between two places on the equator whose longitudes differ by 25°.
  5. Town M is at 30°E, and town N is at 15°W. When it is 3:00 p.m. at M, what time is it at N?

Answers: 1. 6400 cos 45° ≈ 4525 km. 2. The difference in latitude is 25°, so the distance is (25/360) × 2 × (22/7) × 6400 ≈ 2794 km. 3. The difference in longitude is 30°, so the distance is (30/360) × 2 × (22/7) × 4525 ≈ 2370 km. 4. 60 × 25 = 1500 nm. 5. The difference in longitude is 45°, so the time difference is 180 minutes (3 hours). N is west, so it is 12:00 noon there.

Conclusion

Latitude measures how far north or south of the equator a place is, and longitude how far east or west of Greenwich. Along a meridian (a great circle), distance = (θ/360) × 2πR, using the difference in latitude. Along a parallel of latitude α (a small circle of radius R cos α), distance = (θ/360) × 2πR cos α, using the difference in longitude. In nautical miles, these become 60θ and 60θ cos α. Each degree of longitude makes 4 minutes of time difference, with the east ahead, and for places on opposite meridians the route over the pole is the shorter one.

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