Area of Spherical Triangle-1

Here is a presentation for detailed explanation on how to compute area of spherical triangle given aperture angles

One can easily compute the interior angles of spherical triangle ABC using cosine formula

$A=\cos^{-1}\left(\frac{\cos50^\circ-\cos80^\circ\cos100^\circ}{\sin80^\circ\sin100^\circ}\right)=0.803955374$

$B=\cos^{-1}\left(\frac{\cos80^\circ-\cos50^\circ\cos100^\circ}{\sin50^\circ\sin100^\circ}\right)=1.183016058$

$C=\cos^{-1}\left(\frac{\cos100^\circ-\cos50^\circ\cos80^\circ}{\sin80^\circ\sin80^\circ}\right)=1.958576595$

hence area of spherical triangle ABC

$=(A+B+C-\pi)R^2=(0.803955374+1.183016058+1.958576595-\pi)15^2=180.8899595\ cm^2$

Using the cosine rule from spherical trigonometry, which is as follows. For a spherical triangle with sides $a, b, c$ (which are the angles the sides subtend at the center of the sphere), we have the following rule:

$\cos a = \cos b \cos c + \sin b \sin c \cos A$

$\cos b = \cos a \cos c + \sin a \sin c \cos B$

$\cos c = \cos a \cos b + \sin a \sin b \cos C$

Where $A, B, C$ are the spherical angles at the three vertices of the spherical triangle, with angle $A$ between sides $b$ and $c$ , and so on. The above equations determine the desired angles $A, B , C$ . What remains is to use the formula for the area of a spherical triangle, which is,

$\text{Area} = R^2 (A + B + C - \pi )$

Applying the cosine rule above with $a = 50^{\circ} , b = 80^{\circ}, c = 100^{\circ}$ , we obtain (listed in radians).

$A = 0.803955375 , B = 1.183016058, C = 1.958576595$ . Therefore,

$\text{Area} = (15)^2 ( 0.803955375 +1.183016058 + 1.958576595 - 3.141592654 ) = 180.8899593$