By selecting three points in the UHP, a hyperbolic triangle appears. The areas above two of the circular arcs minus the area of the third circular arc gives the area of the hyperbolic triangle showing that the area is pi minus the sum of the angles.
In this demo it is important to state that the result is true only if the point A is between B and C and below the circular arc from B to C. Under those circumstances the area above the red arc plus the area above the blue arc minus the area above the yellow arc equals the area of the hyperbolic triangle with those three arcs. This geometric demonstration is related to the sum of the angles of the triangles, so the hyperbolic area of the triangle is the ? - sum of the angles of the triangle.