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 need to show that if a place has (latitude, longitude) = $(p,q)$ then its coordinates are $$(R \cos p \cos q, R \cos p \sin q, R \sin p).$$

 Steps in the calculation We need the distance $l$ on the surface between A and B. First calculate the three-dimensional coordinates of A and B from the latitude and longitude of the two points. Then calculate the distance $2d$ (imagine a tunnel straight through the earth from A to B) using Pythagoras Theorem. Use $R$, the radius of the Earth, and $d$ to find the angle $\theta$ radians where $$\sin \theta = {d\over R}.$$ Calculate the arc length $l$.