Interior angle sum of polygons (incl. crossed): a general formula

The interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is given by the simple and useful formula S = 180(n-2k) where n is the number of vertices and k = 0, 1, 2, 3. ... represents the number of total revolutions of 360o one undergoes walking around the perimeter of the polygon, and turning at each vertex, until facing in the same direction one started off from. In other words (or put differently), 360k represents the sum of all the exterior angles. For example, for ordinary convex and concave polygons k = 1, since the exterior angle sum = 360o and one undergoes only one full revolution walking around the perimeter.

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Drag vertices A, B, C, D or E. What do you notice about the Angle Sum of the star pentagon in the left column? Can you explain (prove) it?

Though there are several different ways of proving that the interior angle sum of a star pentagon like this is 180o, the above formula gives the result immediately since it's easy to see that k = 2 by rotating a pencil by the exterior angle at each vertex in order, and counting the total number of full revolutions it undergoes. Thus, S = 180(5-2x2) = 180o.

Can you now use the general formula to find the interior angle sum of the following polygons?

More crossed polygons

1) Click on the link below to check the interior angle sum for a crossed septagon like ABCDEFG. Then drag the septagon into other shapes and see if you can correctly predict or explain the shown interior angle sum.

Crossed septagon

2) Click on the link below to check the interior angle sum of a crossed octagon like HIJKLMNO. Then drag the octagon into other shapes and see if you can correctly predict or explain the shown interior angle sum.

Crossed octagon

3) Click on the button below to check the interior angle sum for a crossed quadrilateral like PQRS. Then drag the quadrilateral into the shape of a convex or concave quadrilateral, and then back again into a crossed shape. Can you explain what you observe?

Crossed quadrilateral

Note: My book Rethinking Proof has a learning activity with a worksheet and associated sketch in relation to the defining of 'interior angles' in terms of the concept of 'directed angles' so as to extend it in a consistent way to the interior angle sum of a crossed quadrilateral, and crossed polygons in general.

Also read my article Stars: A second look for a short proof of the formula, and for more examples, see Problem 15 on pp. 49-50 and the Solutions on pp. 135-136 of my book Some Adventures in Euclidean Geometry.

An interactive learning activity that starts with Turtle Geometry that provides a splendid conceptual background to the crucial idea of 'turning angle', and eventually guiding students towards the general interior angle formula is given at: Investigating a general formula for the interior angle sum of polygons.

Michael de Villiers, modified 4 February 2012, created with GeoGebra