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 (http://planetmath.org/AngleMeasure) of each angle of a biangle. Then the biangle covers (http://planetmath.org/Cover) ${\displaystyle \frac{\theta }{2\pi }}$ 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.

Title	biangle
Canonical name	Biangle
Date of creation	2013-03-22 17:06:10
Last modified on	2013-03-22 17:06:10
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	8
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 51M10
Classification	msc 51-00