[Math] Computing surface area/volume of spherical pyramid

areageometryspheresvolume

spherical pyramids1
I want to compute surface area/volume of given 'spherical pyramid' (I don't know whether it is a legit name). What are the answers when $\angle AOB=\angle BOC=\angle AOC = 60 ^\circ$ and $\angle AOB=\angle BOC=\angle AOC = 45 ^\circ$? If there is no geometrical solution, I want to know how to solve in computational tools.

Best Answer

The surface area of the spherical triangle $S_{ABC}=R^2E$ where $R$ is the radius and $E$ is the spherical excess which can be solved by L'Huilier's Theorem.

The volume $V$ can therefore be given by the area ratio of the sphere: \begin{align} V&=V_{sphere}{S_{ABC}\over S_{sphere}}\\ &=\frac{4}{3}\pi R^3{R^2E\over 4\pi R^{2}}\\ &={R^3E\over 3} \end{align}

When $\angle AOB=\angle BOC=\angle AOC = 60 ^\circ$, we have $\tan\tfrac{1}{4}E = (\tan\tfrac{\pi}{12})^{3/2}$, and when $\angle AOB=\angle BOC=\angle AOC = 45 ^\circ$, we have $\tan\tfrac{1}{4}E = \sqrt{(\tan\tfrac{3\pi}{16})(\tan\tfrac{\pi}{16})^{3}}$.

Related Question