Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am asked to calculate the area of a spherical triangle of points $(0,0,1),(\frac{1}{\sqrt2},0, \frac{1}{\sqrt2})$ and $(0,1,0)$.

I know I will have to use Gauss Bonnet formula , after having found the internal angles but since we are given no formula to find them I managed to find by visual analysis just one angle of $90$ degrees.

Help would be appreciated.


share|cite|improve this question
up vote 4 down vote accepted

Think of the $y$ axis as the north-south axis, with $(0,1,0)$ the north pole. Observe that both $(1,0,0)$ and $(1/\sqrt{2},0,1/\sqrt{2})$ lie on the equator (given by $y=0$ on the sphere).

This means that the triangle's angle at the vertex $(0,1,0)$ is $45$ degrees. Since the great circle connecting $(0,1,0)$ and any point on the equator is perpendicular to the equator, the other two vertices of the triangle have angle $90$ degrees.

So the triangle takes up an eighth of the northern hemisphere, thus a sixteenth of the area of the entire sphere: $\frac{4\pi}{16}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.