### How Many Geometries Are There?

An account of how axioms underpin geometry and how by changing one axiom we get an entirely different geometry.

### When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

### Pythagoras on a Sphere

Prove Pythagoras' Theorem for right-angled spherical triangles.

# Flight Path

##### Stage: 5 Challenge Level:

You only need elementary trigonometry to do this question if you use the guidance given in the hints.

London is at longitude $0^o$ and latitude $51.5^o$ North and Sydney at longitude $151^o$ East and latitude $34^o$ South. Taking the earth to be a sphere with radius 6378 kilometres, calculate the distance between London and Sydney.

If the flight path is the shortest route at an altitude of 6 kilometres calculate the distance along the flight path.