# Non-Euclidean area defined by three points on a sphere

1. Jan 4, 2016

### 24forChromium

A sphere with radius "r" has three points on its surface, the points are A, B, and C and are labelled (xa, ya, za) and so on.

What is the general formula to calculate the area on the surface of the sphere defined by these points?

2. Jan 5, 2016

### Samy_A

http://mathworld.wolfram.com/SphericalTriangle.html

Last edited: Jan 5, 2016
3. Jan 5, 2016

### 24forChromium

This is only applicable in the cases where the arcs between the points form parts of "great circles". I need an equation that is applicable to any three non-collinear points.

4. Jan 5, 2016

### WWGD

Why not use the first fundamental form?

5. Jan 5, 2016

### 24forChromium

Please do elaborate. What is the "first fundamental form"? To give you an idea, basic calculus and 3D-vectors is all I can do. (Of course it is also plausible that you are talking about something that I am capable of but have not heard of)

6. Jan 5, 2016

### WWGD

It is ultimately advanced calculus, multivariable calculus, e.g.:

https://en.wikipedia.org/wiki/First_fundamental_form

EDIT: A worked example:
http://math.ucr.edu/~res/math138A/firstform.pdf

7. Jan 5, 2016

8. Jan 5, 2016

### WWGD

Only one I can think of at the moment, let me see if I can think of another one.

9. Jan 5, 2016

### micromass

I don't get it: through any two points on a circle there is a great circle. I don't see how you would give three points that would give rise to a triangle without using great arcs.

10. Jan 5, 2016

### 24forChromium

You are right. Sorry for making a fuzz over nothing, everyone.

11. Jan 6, 2016

### suremarc

I think he means that it only applies for degenerate triangles.

OP: $A=R^2E$ works for all spherical triangles. Look up Girard's Theorem for a proof