the area of a hyperbolic triangle
How do you find the area of a triangle in the hyperbolic plane? In Euclid you would multiply base by height and halve it — but in hyperbolic geometry there is something far stranger and simpler. The area of a hyperbolic triangle depends ONLY on its angles, not on the lengths of its sides at all. Specifically, area = k^2 * (pi - A - B - C), where A, B, C are the three angles in radians and k is the absolute curvature constant of the plane. The bracketed quantity (pi minus the angle sum) is the angle defect; so area is just the defect times a fixed constant.
Read off the immediate consequences. First, every hyperbolic triangle has a defect between 0 and pi, so its area is strictly BOUNDED — there is a largest possible triangle, the 'ideal' triangle whose vertices all sit at infinity (all angles 0), with area exactly k^2 * pi. You cannot make a hyperbolic triangle of arbitrarily large area, no matter how far apart you push the vertices. Second, since the angles alone fix the area, two triangles with the same three angles must have the same area; combined with the angle-angle-angle data, this forces them to be CONGRUENT. There is no scaling up a shape — similar triangles are automatically equal.
This is the cleanest possible illustration of how non-Euclidean geometry differs in kind, not just degree. The proof goes back through the additivity of defect: a triangle splits into pieces whose defects add, and the same holds for areas, so defect and area are proportional, and chasing the ideal-triangle case pins the constant. The spherical twin of this result is Girard's theorem, where area is proportional to the angle EXCESS instead. Both are special cases of the Gauss-Bonnet relation, though the differential-geometric why belongs to a later field.
Take k = 1 and a triangle with angles A = 60, B = 60, C = 60 degrees, i.e. pi/3 each. The angle sum is pi, so the defect is pi - pi = 0 and the area is 0 — but a genuine hyperbolic triangle can never have all angles 60 degrees; that sum forces a degenerate point. A real one with angles 30, 30, 30 degrees (pi/6 each) has defect pi - pi/2 = pi/2, so area = pi/2, about 1.571.
Area = k^2 (pi - A - B - C): determined by angles alone, bounded above by k^2 pi for the ideal triangle.
The startling part is the absence of side lengths: in hyperbolic geometry equal angles force equal area and hence congruence, so similar-but-larger triangles simply do not exist — the AAA criterion gives congruence here, not just similarity.