E4.6 - Reasoning with irregular polygons

Angle sums without equal angles

Can a six-sided polygon have a $270^\circ$ interior angle while its other corners are all right angles? An inward corner makes that possible. The total depends on the side count, not on all the angles being equal.

The interior sum $(n-2)\times180^\circ$ also holds for irregular simple polygons. For a convex one, the fan from one vertex works exactly as before. A concave polygon can also be divided into $n-2$ triangles, but you must choose diagonals that remain inside: an arbitrary vertex fan may run outside the shape. At an inward corner, count the full reflex interior angle, not the small angle in the notch.

[DIAGRAM: asset_name: Convex and concave interior angles; asset_slug: irregular_interior_sums; caption: The inward corner contributes its reflex interior angle.; description: Three exact polygon panels show an irregular convex hexagon with interior angles 90, 135, 135, 90 and 135 degrees known and its sixth angle x; an L-shaped concave hexagon with five right angles and its reflex inward angle y; and a concave pentagon with two 90 degree corners, two 60 degree corners and a reflex inward angle z. The reflex arcs run inside the polygon, around the outside of the notch.]

Keep six sides but change the shape to a concave one. The total stays $720^\circ$; the inward angle must be counted reflex.

Count the sides before deciding the total in the new concave shape.

Interior and exterior relationships

Exterior angles need a consistent convention. For a convex polygon, walking anticlockwise round the boundary makes a left turn at every corner. Those positive turns total $360^\circ$, even when they are unequal.

At a concave corner you turn the other way. If anticlockwise left turns are positive, a right turn is negative. Using these signed exterior turns keeps the total $360^\circ$ for a simple polygon walked anticlockwise. Adding the positive magnitudes of every small outside angle would lose the directions and generally give the wrong total.

[DIAGRAM: asset_name: Unequal and negative exterior turns; asset_slug: irregular_exterior_turns; caption: A right turn at a concave vertex subtracts from the total left turn.; description: Three anticlockwise pentagon walks show exterior turns outside the vertices. The convex example has 80, 50, 95 and 75 degrees plus unknown e. A concave example has turns 90, 90, 135, unknown e at its inward corner, and 135 degrees. A fresh concave example has turns 90, 90, 110, unknown e at the inward corner, and 110 degrees. Positive left-turn arcs and negative right-turn arcs have distinct pale blue and clay colours; arrowheads indicate the travel direction.]