Sum of interior angles of a rectilinear

Question

For a Rectilinear (concave polygon having all sides parallel to either X or Y axis), how do we find the "sum of all the interior angles" given total number of convex corners (corner whose internal angle is 90 degrees).

For example - what will be the sum of all the interior angles of a rectilinear with 10 convex corners.

Any help is much appreciated.

Answer 1

The number of convex corners determines the number of concave corners. We can consider it as a rectangle with smaller rectangles cut out of some corner.

• Starting with a rectangle, we have 4 convex corners and 0 concave corners.
• Each move of removing a smaller rectangle increases the number of convex and concave corners by 1 each.

Consequently, the number of concave corners is $(\#$ convex corners$) - 4$. Now use what you know about the number of degrees in convex and concave corners.