I am building a custom arbour with a semi-circular arch. I wish to calculate how much of an arc I can get out of a given width of lumber. My arch has a 45" inner diameter and is 3 1/2" thick (same thickness as the 4x4 uprights it will sit on) for an outer diameter of 52". If I start with a 2"x10" piece of lumber, it is possible to get a full 90 degree arc out of it. This uses up 36" of wood. But what if I used a 2"x8" piece of lumber? Could I get a smaller degree arc out of that? How long would the lumber have to be? Conversely, what is the maximum angle of arc I could get out of a 2x8? Ultimately, I want to optimize the cost of the arch. 2x10 is 1.89/ft, 2x8 is 1.23/ft.