Not for those who don't remember their trigonometry:

My father recently built a gazebo. Just for fun, he did it with ten
sides, rather than the traditional six or eight. He gave the roof a
5/12 pitch (22.6 degree angle from horizontal). Thus, the roof was
comprised of 10 triangular wedges. He sheathed the roof with planks
forming concentric ten-sided rings around the center.

At what angles did he have to miter the planks to get them to fit
perfectly?

If the roof was flat (zero pitch) like a ten-sided deck, the miter
would have been 36 degrees from perpendicular, with a zero degree tilt
(vertical cut). If the roof was infinitely steep (like building the
walls of a ten-sided tower), he would have to have cut the boards with
a zero degree miter (perfectly perpendicular cross-cut) with a 36
degree tilt from vertical.

What would the formula be for N sides with a roof pitch of A degrees?

Josh