Example

Both Libreville (Gabon) and Kismanyo (Somalia) are on the Equator on opposite sides of the African continent. Libreville is at ${9.27^{\circ}}$ E and Kismanyo is at ${42.32^{\circ}}$ E.

How far is it from Libreville to Kismanyo travelling along the Equator, correct to the nearest km?

Solution

The Equator is a great circle, with a radius of 6400 km.

The angle between the longitude of Libreville and that of Kismanyo is ${42.32^{\circ}}$ ? ${9.27^{\circ}}$ = ${33.05^{\circ}}$.

(We find the difference since both places have the same longitude direction (E))

From Libreville to Kismanyo

\begin{align*} l &=\frac{r\pi}{180^{\circ}}\times\theta \\ &=\frac{6400\pi}{180^{\circ}}\times 33.05^{\circ}\\ &=3691.720\ldots \\ &\approx 3692\text{ km} \end{align*}

Example

Both the Galapagos Islands and the island of Naura are on the Equator, but the Galapagos Islands are at ${90.30^{\circ}}$ W whereas the island of Nauru is at ${166.56^{\circ}}$ E.

How far is it from the Galapagos Islands to Nauru travelling over the Pacific ocean along the Equator, correct to the nearest km ?

Solution

The angle between the longitude of the Galapagos Islands and that of Nauru is ${90.30^{\circ}} + {166.56^{\circ}} = {256.86^{\circ}}$. We find the sum since these places have different longitude directions, but this is the major arc, and the minor arc will be ${360^{\circ}} - {256.86^{\circ}} = {103.14^{\circ}}$.

We could also find the angle between these two places by recognising that both are close to ${180^{\circ}}$E/W. We could find the angle between the Galapagos Islands and ${180^{\circ}}$E/W, the island of Nauru and ${180^{\circ}}$E/W, and then add these two angles together.

Angle between Galapagos Islands and ${180^{\circ}}$E/W = ${180^{\circ}}$ - ${90.30^{\circ}}$ = ${89.70^{\circ}}$.

Angel between Nauru island and ${180^{\circ}}$E/W = ${180^{\circ}}$ - ${166.56^{\circ}}$ = ${13.44^{\circ}}$

Total angle between Galapagos Islands and Nauru = ${89.70^{\circ}}$ + ${13.44^{\circ}}$ = ${103.14^{\circ}}$

From the Galapagos Islands to Nauru

\begin{align*} l &=\frac{r\pi}{180^{\circ}}\times\theta \\ &=\frac{6400\pi}{180^{\circ}}\times 103.14^{\circ} \\ &=11520.848\ldots \\ &\approx 11521\text{ km} \end{align*}