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Distances along Small Circles is taught in the note for The Earth as a Sphere. Read it below and look for the section on this topic.

The Earth as a Sphere

12 min read

Latitude and longitude

The Earth is nearly a sphere of radius about 6 370 km. Positions are described by latitude, the angle north or south of the Equator (0° to 90°), and longitude, the angle east or west of the Greenwich meridian (0° to 180°). Lines of longitude are great circles passing through both poles. The Equator is also a great circle; the other parallels of latitude are small circles.

The layers of the EarthA cross-section of the Earth. A thin solid crust lies on top, then a thick mantle of hot rock that flows very slowly, then a liquid outer core of iron and nickel, and a solid inner core at the centre which is very hot but kept solid by pressure.crust (5–70 km)mantle (2 900 km thick)outer core: liquid (2 200 km)inner core: solid (radius 1 200 km)
Why we have seasonsThe Earth circles the Sun once a year with its axis tilted by about 23.5 degrees and always pointing the same way. When the northern hemisphere leans towards the Sun it has summer, longer days and more direct light. Six months later it leans away and has winter. Near the equator, as in Tanzania, the change is small.SunJune: northern summerSeptember: equinoxDecember: northern winterMarch: equinox

Distances along a great circle

The distance along a great circle depends on the angle at the centre of the Earth. One degree on a great circle is 60 nautical miles. A nautical mile is about 1.852 km. One knot is a speed of one nautical mile per hour.

θ is the angle at the centre, R = 6 370 km.

Distances along a small circle

A parallel of latitude φ is a small circle of radius R cos φ, which is smaller than the Equator. The distance between two places on the same parallel with a difference in longitude Δλ is:

Cos φ shrinks the distance as you move away from the Equator.
Worked exampleDistance along a parallel

Find the distance between two places on latitude 60° N whose longitudes differ by 30°. (R = 6 370 km)

  1. Radius of the small circle.
  2. Fraction 30/360 of its circumference.

Answer: About 1 668 km.

The distance falls with latitude, to zero at the poles.
Try itDistance on a parallel

Choose the latitude and the difference in longitude.

Check yourself One degree of arc on a great circle is
Check yourself The radius of the parallel of latitude 60° is
Check yourself A speed of 1 knot is

Key terms

Latitude
Angle north or south of the Equator.
Longitude
Angle east or west of Greenwich.
Great circle
A circle whose centre is the centre of the Earth.
Nautical mile
The distance along a great circle for 1 minute of arc.

Summary

  • Latitude and longitude locate any place.
  • Along a great circle 1° is 60 nautical miles.
  • On a parallel the distance is multiplied by cos of the latitude.
  • One knot is one nautical mile per hour.
Distances