Is there a way with PHP GD to draw a star using imagefilledpolygon?
Where would the points be plotted?
I believe it involves sine and cosine because…

How can I relate those points to the center using sine and cosine in GD?
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yes. I suggest you read that manual entry you yourself supplied, as it tells you exactly what you need to do. It even gives an example of a 3-point star, and there’s even an example in the user notes of a 5-sided star.