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LOGO Challenge 9 - Overlapping Polygons

This LOGO challenge starts by looking at 10-sided polygons then generalises the findings to any polygon, putting particular emphasis on external angles

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Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

Two Regular Polygons

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
Two polygons fit together so that the exterior (orange) angle at each end of their shared side is $81^\circ$.
Two irregular polygons
If both shapes now have to be regular polygons, but do not need to be the same, and each polygon can have any number of sides, could the orange angle still be $81^\circ$, and if that is possible how many sides would each polygon have?
Find solutions for when the orange angle is $27^\circ$ and when it is $54^\circ$.

Can you make a conjecture about the connection between the size of the orange angle and the number of sides on each polygon.
If you can, are you able to justify your conjecture?