Biangle 2007-05-19 12:45:09 2007-10-27 04:49:56 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{psfrag} \usepackage{graphicx} \usepackage{amsthm} \usepackage{xypic} In spherical geometry, it is possible to form a polygon with only two sides. Thus, we have the following definition: A \emph{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 \PMlinkname{measure}{AngleMeasure} of each angle of a biangle. Then the biangle \PMlinkname{covers}{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.