# Applying equation for the area of a spherical $n$-gon

I'm currently studying geometry and came across an exercise problem that I can't wrap my head around. I tried looking at another question on this community (Area of spherical polygon) and although it's almost what I'm looking for I'm getting confused regarding the application of the formula.

Problem: Analogously to the planar case, formulate an equation for the area of a spherical $$n$$-gon.

Solution: Similarly to the Euclidean case, divide the $$n$$-gon into $$n - 2$$ triangles and add the associated area formula for each triangle:

$$\text{Area} = \sum_{i}^n \theta_i - (n - 2)\pi$$

I tried using the following example:

but what I'm confused about is that if we were to use the rightmost triangle then we would get something like

$$\left(\frac{\pi}{5} + \frac{3\pi}{5} + \frac{\pi}{5} \right) - \pi$$

which is $$0$$. How should I be applying this formula? Thanks.

• You're applying it to a Euclidean pentagon, or equivalently, an infinitesimally small pentagon on the unit sphere, hence the $0$ area. A spherical convex regular pentagon would have an angle between $108^\circ$ and $180^\circ$. Aug 6, 2020 at 23:48