biangle

In spherical geometry, it is possible to form a polygon with only two sides. Thus, we have the following definition:

A biangle is a two-sided polygon that is strictly contained in one hemisphere of the sphere that is serving as the model for spherical geometry.

Given a biangle, its vertices must be antipodal points, and its two angles must be congruent. Therefore, every biangle is equiangular. Since each side of a biangle is half of a great circle, every biangle is equilateral. Hence, every biangle is regular.

Let $\theta$ be the radian measure of each angle of a biangle. Then the biangle covers $\displaystyle \frac{\theta}{2\pi}$ of the sphere. Since the area of the sphere is $4\pi$ , the area of the biangle is $2\theta$ . Note that this equals the angle sum of the biangle.