Angle excess

Angle excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. For instance, a plane triangle has an angle sum of 180°; an octant is a spherical triangle with three right angles, so its angle sum is 270°, and its angle excess is 90°. The angle excess of any polygon on a sphere is proportional to the polygon's area, with the proportionality constant being the square of the sphere's radius.

In surveying, one checks whether the angles and distances form a closed polygon, and by how much it is off. If the area is sufficiently large, the polygon will not close no matter how accurately measured if it is calculated on a plane. The area of a polygon whose angle excess is 1 second of arc, which is the precision (though not necessarily the accuracy) of surveying, is 393 square kilometers, or about 20 kilometers square.

Angle deficit is defined similarly on a pseudosphere, and is likewise proportional to area in hyperbolic geometry.