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Guy, We did have a miscommunication. In my diagram below, which is the base of a lamp with 5 sides, I took "diameter" to mean the diameter of the red circle where you meant the diameter of the blue circle. Suppose the diameter of this circle is $d$ then in my first diagram $CD = d/2.$ Then $\frac{DB}{CD} = tan\left(\frac{180}{n}\right)$ and hence \[L = d\;tan\left(\frac{180}{n}\right).\] Harley  


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