# Area of hyperbolic triangle

1. Aug 21, 2009

### skippy1729

Given a triangle on a hyperbolic surface with all angles and edge length known the area is given by R^2 x(PI - a - b - c), where a, b and c are the angles and R is the radius of curvature of the surface. What if you don't know R?

Same question for a triangle on a spherical surface where R is unknown.

Equivalent question: How do you measure R using LOCAL length and angle measurements? Assume that you know only that the surface has a constant curvature.

Thanks, Skippy

2. Aug 23, 2009

### Reb

If I understand your question, you can find $$R=1/\sqrt{|K|}$$ by means of the Gaussian curvature $$K={\rm det}(II)/{\rm det}(I)$$, $$I$$ and $$II$$ being the first and second fundamental forms of the surface.

However, you can always invoke the Gauss-Bonnet formula:

http://mathworld.wolfram.com/Gauss-BonnetFormula.html

Especially if you restrict your study to geodesic triangles, you can derive the formulas for their areas.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook