Any-sided shapes with angles

A circle used distance; a box used edges. A regular polygon — triangle, pentagon, hexagon — uses the ANGLE around the center. If you know the direction from the middle to a pixel, you know which of the polygon’s flat sides that pixel faces, and how far it is from that side. atan(pos.x, pos.y) gives you that angle, and one compact formula turns it into a distance to the nearest edge.

The magic line is d = cos(floor(0.5 + a/r) * r - a) * length(pos), where r is one slice of the circle (2π / number of sides). You do not need to memorize it — the thing to see is that a single sides number controls the whole shape. Change it and the triangle becomes a pentagon becomes a hexagon.

Change sides to 3 and you get a triangle; 5 a pentagon; 8 an octagon; push it high and it rounds back into a circle. This is how shaders draw gems, stars, badges and UI icons — the same formula, one parameter.