Does It Close?

Where it usually goes wrong

Stepping a compass round a circle to make a regular polygon gets remembered as a trick tied to one shape — open it to the radius, get a hexagon — as if 6 were a fact about circles. It is a fact about DIVISION: a step of θ degrees closes the circle after n steps exactly when θ × n = 360, which is true for a triangle at 120°, a square at 90°, a pentagon at 72°, an octagon at 45°, and every other regular polygon a compass can step out, each for the same one reason.

Try this

• Drag until the step count reads a whole number of steps that also lands exactly on the first mark — a pentagon, a square, an octagon. Read θ times that count: it is 360 every time.
• Drag to an angle that does not divide 360 evenly. The last step lands short of the first mark, with a visible gap — there is no whole number of these steps that gets all the way round.
• Drag past that point instead. Now the steps overshoot: the last one has already walked past the start and the figure overlaps itself.

Where this goes next

Joining every other mark of a hexagon makes a triangle, and both are on the inscribe-in-a-circle worksheet — a square uses a different method entirely (two perpendicular diameters), because no single step size can ever divide a circle into four this way.