### 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:

 A model to demonstrate latitude and longitude Cut 2 discs of cardboard. Cut a radial slit in each so that they slot together as shown. Mark your chosen cities and their angles of latitude. In the illustration London at $51.5^o$ North and Sydney at $34^o$ South are shown. Cut off a small segment at the South Pole so that the model stands up. Adjust the angle between the two planes in the model to show the difference in their angles of longitude. Take the origin at the centre of the earth, the x-axis through the Greenwich Meridian at the equator, the y-axis through longitude $90^o$ East at the equator and the z-axis through the North Pole. London is approximately on the Greenwich Meridian so the 3-D coordinates of London are $(R\cos 51.5^o, 0, R\sin 51.5^o$).

As an extension to this question you might like to calculate the distance between Cape Town ($18^o$E, $34^o$S) and Sydney ($151^o$E, $34^o$S) along the great circle and find how much shorter it is than the distance around the line of latitude $34^o$S.