Pierced polygons

Task: Investigate the sum of the internal angles of a regular polygon that has another regular polygon removed from its centre.

Prompts from the teacher could be:

1. Set up a table for regular polygons that gives the sum of internal angles and the sum of the 'outside' reflex angles. (This is different to the sum of external angles)
2. Try a few pairs of polygons that make up a pierced polygon (eg a square with an equilateral triangle cut out and vice versa). What do you notice?
3. Divide your sum of all the internal and outside reflex angles by 180°. What do you notice?

A prompt from the teacher could be to suggest approaching this problem of internal angles the same way that the class may have derived the rule for the sum of internal angles of a polygon – ie, to divide the shape into triangles, each being 180°.
Students may be encouraged to explore whether their rule would hold for irregular pierced polygons.

Students may be encouraged to explore whether their rule would hold for irregular pierced polygons.