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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.

The question is simply asking:

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:

enter image description here

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.

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  • $\begingroup$ 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 $180^\circ$. $\endgroup$
    – mr_e_man
    Aug 6, 2020 at 23:48

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