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