Angle excess

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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 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 1 second of arc, which is the precision (though not necessarily the accuracy) of surveying, is 393 square kilometres, or about 20 kilometres square.

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

See also