You may also like

problem icon

How Many Geometries Are There?

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

problem icon

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.

problem icon

Pythagoras on a Sphere

Prove Pythagoras' Theorem for right-angled spherical triangles.

Flight Path

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

section
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 $l$.