The Poincare half-plane model is conformal, which means that hyperbolic angles in the Poincare half-plane model are exactly the same as the Euclidean angles -- with the angles between two intersecting circles being the angle between their tangent lines at the point of intersection. A hyperbolic triangle is just three points connected by (hyperbolic) line segments. Despite all these similarities, hyperbolic triangles are quite different from Euclidean triangles.
Since the hyperbolic line segments are (usually) curved, the angles of a hyperbolic triangle add up to strictly less than 180 degrees. Since a quadrilateral can always be cut into two triangles, a quadrilateral must have its angles add up to less than 360 degrees, so in hyperbolic geometry there are no squares, which makes defining area in terms of square units difficult. It turns out, however, that there is a unique function (up to multiplication by a scale factor) that satisfies the usual area axioms:
For a triangle, this function is the defect, defined as
where the angles alpha, beta, and gamma are measured in radians.
In addition to the smaller angle sum, there are two other major differences between hyperbolic triangles and Euclidean triangles:
In the applet you will have a red point, a blue point, and a black point. The points are connected by (hyperbolic) line segments to make a triangle. The angles of the triangle are listed below the edge of the half-plane. They are color coded, so the angle at the red point is given in red, etc. Below these angles are the sum of the three angles and the area of the triangle. Click your mouse on a point and drag it (while holding the mouse button down) to move the point. The triangle will follow the point as in the other applets. (Bug warning: Sometimes when the window is covered and then uncovered by other windows on your computer monitor the applet doesn't redraw itself completely. If you click on a point and move it, or if you minimize and then restore the window, the applet will redraw itself properly again.)
Things to try
Click here to launch applet. (It will open a new window.)