Calculate the Angular Sum of an N-gon
Calculator for the angular sum of a polygon or n-gon, a two-dimensional closed shape with straight lines. For every shape with n vertices and straight edges, the sum of the inner angles is equal. Best known is this from the triangle with its angular sum being always 180 degrees. For a quadrilateral, this is 360 degrees. The average angle is the arithmetic mean of the n-gon's angles. For a regular n-gon, it is the exact size of each angle.
Please enter for the number of vertices a natural number ≥3, the angular sum and the average angle will be calculated. This calculation only applies to Euclidean geometry, i.e. to shapes on a straight plane. In non-Euclidean geometry, other angle sums occur, smaller ones on a saddle-shaped surface, larger ones on a sphere.
The formulas for calculating the sum of the angles in a polygon are:
Angular sum = n * 180° − 360°
Average angle = 180° − ( 360° / n )
The average angle is at least 60 degrees (this is the value for a triangle) and approaches 180 degrees for large n, but is always lower.
With each additional vertex, the sum of the angles increases by 180 degrees. You can also see an n-gon as an n+1-gon by considering any point in the middle of one of the sides as a vertex with an angle of 180 degrees.
This angular sum also applies to concave polygons, i.e. those where one or more corners point inwards. However, it does not apply to crossed polygons or to non-Euclidean shapes. Non-Euclidean shapes are those that do not lie on a plane but on a curved surface. An example of such a shape is the spherical triangle, where the angular sum is greater than 180 degrees. On elliptical surfaces, the shapes have larger angular sums, on hyperbolic surfaces, the shapes have smaller angular sums.