Angle excess, also known as spherical excess is the amount by which the sum of the angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s of a polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

 on a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 exceeds the sum of the angles of a polygon with the same number of sides in a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

. For instance, a plane triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 has an angle sum of 180°; an octant

Octant

An octant is one of eight divisions.-Octant in the plane :Traditionally wind direction is given as one of the 8 octants because that is more accurate than merely giving one of the 4 quadrants, and the wind vane typically does not have enough accuracy to bother with more precise indication.-Octant...

 is a spherical triangle with three right angles, so its angle sum is 270°, and its angle excess is 90°. The area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of any polygon on a sphere is proportional

Proportionality (mathematics)

In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

 to the polygon's angle excess, with the proportionality constant being the square of the sphere's radius when the angle excess is given in radians.

In surveying

Surveying

See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

, 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

Accuracy and precision

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

 (though not necessarily the accuracy) of surveying, is 393 square kilometres, or about 20 kilometres square.

Angle deficit is defined similarly in hyperbolic geometry

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

and is likewise proportional to area.