the angle excess
On a flat plane a triangle's angles sum to exactly 180 degrees; in the hyperbolic plane they fall short. The angle excess is the mirror fact for positively curved surfaces like the sphere: there the three angles add up to MORE than 180 degrees, and the excess measures the overshoot. excess = (angle A + angle B + angle C) - 180 degrees, always a positive number on the sphere. It is the exact sign-flipped twin of the hyperbolic angle defect.
The picture: draw a triangle on an orange with three great-circle arcs. Because the surface bulges outward, the corners are pushed wider than on a flat page, so the angles total beyond 180 degrees. A small triangle (relative to the sphere's size) bulges only slightly and has a tiny excess — which is why a triangle drawn on your desk, microscopic against the curvature of the Earth, seems to obey the flat 180-degree rule. The bigger the triangle, the bigger the excess: a triangle covering a whole octant of the globe has three right angles and an excess of a full 90 degrees.
The excess matters because, exactly as with hyperbolic defect, it MEASURES AREA. Girard's theorem states that a spherical triangle's area equals R^2 times its excess (with the excess in radians) — angles alone determine area, no side lengths required. This makes the excess a practical surveying and navigation tool, and it ties spherical, flat, and hyperbolic geometry into one family: positive excess, zero, and negative (the defect) correspond to positive, zero, and negative curvature. As ever, the deeper reason via curvature belongs to differential geometry; here it is a clean angle-arithmetic fact.
A spherical triangle on a globe has angles 95, 100, and 92 degrees. Their sum is 287 degrees, so the angle excess is 287 - 180 = 107 degrees, which in radians is about 1.868. By Girard's theorem its area is R^2 times 1.868; on Earth (R about 6371 km) that is a substantial chunk of the planet's surface.
Excess = angle sum minus 180 degrees; positive on the sphere, and exactly proportional to area (Girard).
Excess (sphere, sum above 180) and defect (hyperbolic, sum below 180) are opposite-signed versions of one idea; both are exactly 0 only in flat Euclidean geometry. Tiny triangles have negligible excess, which is why everyday flat geometry works so well.