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Distances along meridians

Example

Paris (France) is at ${48.67^{\circ}}$ N and ${2.33^{\circ}}$ E.

How far is it from Paris to the North Pole and the Equator travelling along the meridian, correct to the nearest km ?

Solution

Each meridian is a great circle, with a radius of 6400 km.

The angle between the latitude of Paris and that of the Equator is ${48.67^{\circ}}$.

The angle between the latitude of Paris and the North Pole is ${90.00^{\circ}}$ - ${48.67^{\circ}}$ = ${41.33^{\circ}}$.

From Paris to the Equator From Paris to the North Pole
\begin{align*}&l=\dfrac{r\pi}{180^{\circ}}\times\theta\\ &=\dfrac{6400\pi}{180^{\circ}}\times 48.67^{\circ}\\ &=5436.491\ldots\\ &\approx 5436\text{ km}\end{align*} \begin{align*}&l=\dfrac{r\pi}{180^{\circ}}\times\theta\\ &=\dfrac{6400\pi}{180^{\circ}}\times 41.33^{\circ}\\ &=4616.605\ldots\\ &\approx 4617\text{ km}\end{align*}

Note: The metre was originally defined as one ten-millionth of the distance from the North Pole to the Equator travelling along the meridian through Paris. Owing to some errors in estimating the shape of the earth, the defined metre was about one-fifth of a millimetre shorter than the actual distance, meaning that the actual circumference of the earth through the poles is 40 007 863 m rather than the expected 40 000 000 m.)

Example

Melbourne (Victoria) is at ${37.82^{\circ}}$ S and ${144.97^{\circ}}$ E.

How far is it from Melbourne to the South Pole, the Equator and the North Pole travelling along the meridian, correct to the nearest km ?

Solution

Each meridian is a great circle, with a radius of 6400 km.

From Melbourne to the Equator

The angle between the latitude of Melbourne and that of the Equator is ${37.82^{\circ}}$.

$\begin{aligned}[t] &=\frac{r\pi}{180^{\circ}}\times\theta\\ &=\frac{6400\pi}{180^{\circ}}\times 37.82^{\circ}\\ &=4224.53\ldots\\ &\approx 4225\text{ km} \end{aligned}$

From Melbourne to the South Pole The angle between the latitude of Melbourne and the South Pole is ${90.00^{\circ}}- {37.82^{\circ}} = {52.18^{\circ}}$

\begin{align*} &l=\frac{r\pi}{180^{\circ}}\times\theta\\ &=\frac{6400\pi}{180^{\circ}}\times 52.18^{\circ} \\ &=5828.56\ldots \\ &\approx 5829\text{ km} \end{align*}

From Melbourne to the North Pole The angle between the latitude of Melbourne and the North Pole is ${90.00^{\circ}} + {37.82^{\circ}} = {127.82^{\circ}}$

\begin{align*} &l=\frac{r\pi}{180^{\circ}}\times\theta\\ &=\frac{6400\pi}{180^{\circ}}\times 127.82^{\circ} \\ &=14277.63\ldots \\ &\approx 14278\text{ km} \end{align*}

Distances between two places

Finding the distance between two places on the surface of the Earth implies finding either the great circle distance (the shortest distance between any two points on the surface of a sphere) or the small circle distance (travelling along a parallel of latitude). To complete these calculations, we need to know the location (latitude and longitude) of each place.

There are four types of calculations that can be performed for this kind of problem.

The first two are part of the Further Mathematics course, and the third is a reasonable extension of this, and so the first three will be covered below. The fourth is not part of the Further Mathematics course, but will be covered in the Extension Activity at the end of these notes.

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