Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Area of hyperbolic triangle

  1. Aug 21, 2009 #1
    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. jcsd
  3. Aug 23, 2009 #2


    User Avatar

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

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


    Especially if you restrict your study to geodesic triangles, you can derive the formulas for their areas.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook