the angle defect
Every schoolchild learns that a triangle's three angles add to exactly 180 degrees. That comforting fact is a THEOREM of Euclidean geometry — it depends on the parallel postulate, and it fails in the hyperbolic plane. There, every triangle's angles add to strictly LESS than 180 degrees. The angle defect of a triangle is exactly how much it falls short: defect = 180 degrees - (angle A + angle B + angle C), always a positive number in hyperbolic geometry.
The defect is not a defect in the sense of an error — it is a genuine geometric quantity, and it behaves beautifully. It is ADDITIVE: cut a triangle into two by a line from a vertex, and the two pieces' defects add up to the whole triangle's defect, the way areas add. Small triangles have small defect (a tiny hyperbolic triangle is nearly Euclidean, angle sum just under 180); large triangles have large defect. The defect can approach but never reach a maximum of 180 degrees, attained only by an 'ideal' triangle whose three vertices have run off to infinity, all three angles shrinking to 0.
Its importance is that the defect MEASURES AREA. The area of a hyperbolic triangle is exactly proportional to its angle defect — a clean, exact law with no Euclidean analogue, since in Euclid the defect is always 0 yet areas vary freely. (The spherical mirror-image is the angle EXCESS, where angles overshoot 180 degrees.) Because of this, you can determine a hyperbolic triangle's area just by measuring its three angles, never touching a ruler.
A hyperbolic triangle has angles 50, 60, and 65 degrees. Its angle sum is 175 degrees, so its defect is 180 - 175 = 5 degrees. A larger triangle with angles 30, 40, 50 sums to 120, giving a defect of 60 degrees and hence twelve times the area of the first — area follows defect, not the lengths of the sides.
Defect = 180 degrees minus the angle sum; it is additive and exactly proportional to area in the hyperbolic plane.
In Euclidean geometry the defect is always exactly 0, so the concept is empty there — it is a strictly non-Euclidean idea. Do not confuse the hyperbolic DEFECT (sum below 180) with the spherical EXCESS (sum above 180); they are opposite signs.