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

## Main Question or Discussion Point

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?

Samy_A
Homework Helper
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?
http://mathworld.wolfram.com/SphericalTriangle.html

Last edited:
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.

WWGD
Gold Member
2019 Award
Why not use the first fundamental form?

Why not use the first fundamental form?
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)

WWGD
Gold Member
2019 Award
WWGD
Gold Member
2019 Award
Is that the only way to calculate the area of a triangle on a sphere that may not consist of great circle arcs?
Only one I can think of at the moment, let me see if I can think of another one.

• 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.
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.

Only one I can think of at the moment, let me see if I can think of another one.
You are right. Sorry for making a fuzz over nothing, everyone.

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.
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