- Angle excess
**Angle excess**is the amount by which the sum of theangle s of apolygon on asphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. For instance, a planetriangle has an angle sum of 180°; anoctant is aspherical 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'sarea , with the proportionality constant being the reciprocal of 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 1second of arc , which is the precision (though not necessarily theaccuracy ) of surveying, is 393square kilometre s, or about 20kilometre s square.**Angle deficit**is defined similarly on apseudosphere , and is likewise proportional to area inhyperbolic geometry .**ee also***

defect (geometry)

*spherical trigonometry

*steradian

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