Both Quito and Kampala lie on the Equator. The longitude of Quito is 82°30'W and that of Kampala is 37°30'E. What is the distance from Quito to Kampala? A) along the shortest surface path B) along a direct through the Earth path? Please help me visualize it with a figure. How to do it?


This is hardly an astronomy question, but I like drawing, so here you are:

Earth from above

Longitudes are measured from the Greenwich meridian, so the angle between Kampala and Quito is $$\theta_\mathrm{Q} + \theta_\mathrm{K} = 82.5^\circ + 37.5^\circ = 110^\circ.$$ (remember that $0^\circ30' = 0.5^\circ$). The shortest surface path is along Equator. Since $110^\circ$ is $\frac{110^\circ}{360^\circ} \simeq 0.3$ times the circumference of Earth at Equator, the length of path A (the dashed line) is $$\mathrm{A}:\,\,d = 0.3 \times 40,075\,\mathrm{km} = 12,245\,\mathrm{km}$$

For path B (the solid line), you need a bit more trigonometry. The radius of Earth is $R = 40,075\,\mathrm{km}\,/\,2\pi = 6,378\,\mathrm{km}$. The rest will be left as an exercise.

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  • $\begingroup$ Strange mixture of significant figures. The radius to match the circumference you have used is 6378 km. $\endgroup$ – Rob Jeffries Apr 13 '15 at 9:17
  • $\begingroup$ @RobJeffries: That's because the 6371 km is an "average" over the whole Earth. At the equator, it is 6378 km, as you say. Thanks for pointing that out, I'll fix it. $\endgroup$ – pela Apr 13 '15 at 10:38

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