Triangles on a ball

On a sphere the "straight lines" are great circles — circles whose centre is the centre of the sphere, like the equator. The walk in the first hint makes a triangle with three right angles: its angles add up to $270^\circ$.

Albert Girard found the rule in 1629. On a sphere of radius $R$, a triangle with angles $\alpha, \beta, \gamma$ (in radians) has

$$\alpha + \beta + \gamma - \pi = \frac{\text{area}}{R^2}.$$

Our triangle covers one eighth of the sphere, so its area is $\tfrac18 \cdot 4\pi R^2 = \tfrac{\pi}{2}R^2$, and the excess is $\tfrac{\pi}{2}$ — exactly the extra $90^\circ$.

Pythagoras fails too. All three sides of that triangle are quarter-circles of the same length $c$, so $a^2 + b^2 = 2c^2 \neq c^2$. The spherical version of the theorem is

$$\cos\frac{c}{R} = \cos\frac{a}{R}\,\cos\frac{b}{R},$$

and for small triangles it is very close to $a^2 + b^2 = c^2$ — which is why flat geometry works so well for drawing a field or a town, even though the Earth is round.

The difference between the angle sum and $180^\circ$ measures how curved a surface is. On a saddle-shaped surface — the hyperbolic plane — triangles have angle sums less than $180^\circ$. Gauss, Bolyai, Lobachevsky and Riemann turned these observations into non-Euclidean geometry, and Einstein used curved geometry to describe gravity.