# Computing area of equilateral triangle on a sphere

1. Sep 9, 2012

### tainted

1. The problem statement, all variables and given/known data

Suppose $T$ is an equilateral triangle on the sphere of radius $R = 1$. Let $\alpha$ denote the angle at any of the three vertices’s of the triangle. (Recall that $3\alpha > n$.) Use the result of the last problem on the previous homework and the inclusion - exclusion principle (together with an orange and a knife) to compute the area of $T$ .

2. Relevant equations

The result to the last problem on the previous homework is $A = \alpha2R^2$

3. The attempt at a solution
I assumed that all angle on the equilateral triangle where 90 degrees or $\frac{\pi}{2}$; therefore making the volume equal to 1/8 that of the whole sphere

So I did
$A = \alpha2R^2$ where $A$ is the area of $T$
$A = \frac{\pi}{2}2R^2$
$A = \pi*R^2$ That would be the area of 1/4 of the sphere overall, but because I am taking the area of an equilateral triangle, I took half of that to get
$A = \frac{\pi}{2}R^2$
$A = \frac{\pi}{2}*1$
$A = \frac{\pi}{2}$

Would that be correct? I just kind of picked 90 degrees or $\frac{\pi}{2}$ for $\alpha$, but I assume it could be anything between 60 and up to 90 degrees which would change my answer. How do I know which angle to pick?

Last edited: Sep 9, 2012
2. Sep 9, 2012

### tainted

I am concerned that my answer will be completely different if I used an angle other than 90 degrees. Also I do not know what an orange and knife have to do with this problem.

3. Sep 9, 2012

### Dick

You are supposed to do it for ANY angle using inclusion/exclusion. Look at the pictures here: http://planetmath.org/AreaOfASphericalTriangle.html [Broken] Just look at the pictures, ignore the solution and try to work it out for yourself.

Last edited by a moderator: May 6, 2017