## Extension Activity

### Comparing great circle distance with small circle distance

#### Example

Compare the small circle distance between Capetown (${33.5^{\circ}}$ S, ${18.22^{\circ}}$ E) and Sydney (${33.5^{\circ}}$ S, ${151.13^{\circ}}$ E) with the great circle distance.

Solution

From a previous calculation, the small circle distance between Capetown and Sydney was found to be 12380 km.

For the great circle calculation, both latitudes will have negative values and the absolute difference in longitudes will be ${151.13^{\circ}}$ - ${18.22^{\circ}}$ = ${132.91^{\circ}}$.

In one operation with a graphic calculator,

\begin{align*} d&=\frac{6400 \times \pi}{180} \times \cos^{-1} \big((\sin {-33.5} \times \sin {-33.5} ) + (\cos {-33.5} \times \cos {-33.5} \times \cos {132.91})\big)\\ &= 11138.642\ldots\\ &\approx 11139\text{ km} \end{align*}

From this we can see that the great circle distance is considerably shorter (by 1241 km) than the small circle distance.

If we put this value as the answer to our normal great circle calculation, we get

\begin{align*} 11139 &=\frac{6400\pi}{180^{\circ}}\times \theta^{\circ}\\ \text{Giving} \; \theta &=\frac{11139 \times 180^{\circ}}{6400\pi}\\ \theta &=99.72^{\circ} \end{align*}

So, due to the curvature of the earth at latitude ${33.5^{\circ}}$S, the angle between the radii to Capetown and Sydney is actually just under ${100^{\circ}}$.