Angle sum

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The (interior) angle sum of a planar geometric figure is usually the sum of all interior angles in the figure.

Sum of angles in Euclidean geometry

If the polygon lies in a Euclidean plane , the sum of the angles is given by the formula

given, where stands for the number of corners of the polygon.

Examples

The formula results in the values ​​of the angle sums for triangles , squares and pentagons :

  • for triangles ( ):
  • for squares ( ):
  • for pentagons ( ):

Derivation of the formula

Triangles

To prove the sum of angles in the triangle

That the sum of the interior angles in the triangle is 180 ° follows from the axioms of Euclidean geometry.

General

You can divide a convex corner with the help of a point inside into partial triangles, which then have a total angle of . However, you still have to subtract the full angle around this point, so

Alternatively, one can say that a corner is made up of diagonals that divide the polygon into partial triangles, which is the sum of the angles . This shows the formula.

Sum of angles in non-Euclidean geometry

In a non-Euclidean plane with positive curvature , for example on the surface of a sphere , the sum of the angles is always more than the specified values. In general, the larger the polygon, the greater the deviation. Example: On earth, the triangle formed by the equator , the prime meridian and the 90th degree of longitude has the angle sum 270 °.

In a non-Euclidean plane with negative curvature , for example on a saddle surface , the sum of the angles is always less than the specified values. It can even assume values ​​that are arbitrarily close to 0.

Individual evidence

  1. ^ Translation of the proof from Euclid's "Elements": I.32 to I 31 ( Memento from June 24, 2013 in the Internet Archive ).