Find a spherical triangle with angle sum $5π/3$

Question

Find a spherical triangle with angle sum $5π/3$

I am unsure how to answer this question and would like to be shown how I go about answering?

Answer 1

You can think of a spherical triangle being on the surface of a sphere and having angles that sum greater than $\pi$, or $180^\circ$, but less than $2\pi$, or $360^{\circ}$.

That means you can have two right angles in the triangle. So one possible solution would be to have two sides perpendicular to another (visually this might be like two lines of longitude intersecting the equator on the global map of Earth), and the included angle between those sides having measure $\frac{2\pi}3$ (or $120^{\circ}$). So the three angles of the triangle would be $\frac{\pi}2$, $\frac{\pi}2$, and $\frac{2\pi}3$, since

$$\frac{\pi}2+\frac{\pi}2+\frac{2\pi}3=\frac{5\pi}3.$$

Answer 2

In general, the excess of a spherical triangle (i.e., the difference between the sum of the angles and the "ordinary" pi for planar triangles) is equal to its surface area on the unit sphere. If the sum is 5pi/3 then the area is 2pi/3. The complete sphere is 4pi and a hemisphere is 2pi, so you could use one-third of a hemisphere. This is an elaborate way of arriving at John Molokach's clever solution...