Trigonometry/For Enthusiasts/The Distance from New York to Tokyo

If we assume for simplicity that the Earth is a perfect sphere, it is possible to calculate the shortest distance between any two points on its surface. The shortest path is along the great circle through the two points. A great circle is so called because its radius equals the radius of the sphere, the largest possible radius for a circle on a sphere; it is the intersection of a plane passing through the centre of the sphere and the sphere's surface. The two points and the centre define a unique plane (unless the points are at opposite ends of a diameter of the sphere) hence the great circle through them is unique.

Worked Example: Distance between New York and Tokyo
The Earth

Earth's Radius: 6,371 km

  Lattitude Longitude
Tokyo 35°42′N 139°42′E
New York 40°43′N 74°0′W
Exercise: Promontorium Laplace to Crater Clavius
Crater Clavius

Moon's Radius: 1737 km

  Lattitude Longitude
Promontorium Laplace 46°00′N 25°48′W
Crater Clavius 58°24′S 14°24′W
Exercise: Where you live to Auckland

Use the internet to find the latitude and longitude of where you live and of Auckland (New Zealand) and then work out the distance.

If you live in Auckland (New Zealand), work out the distance to Auckland (California).