I want to make a dome (1/2 sphere) that is made up of 10 metal
"triangular" sections. It will not be a true 1/2 sphere. Each
metal section will be bent in only one direction and will be straight
in the other. The base and each horizontal cross section will be a
Decagon (10 equal sides). Each vertical cross section will be some
sort of a 1/2 circle. I am trying to figure out how to make the 10
"triangles" so that the sides will touch when bent.

If I take 10 isosceles triangles and arrange them so their bases form a
Decagon then bend the points in to touch each it will form a dome, but
the triangles will not touch along their sides, only at the base and
the top. The sides of each triangle need to be curved out. Is there
any way to make this proper curve without trial and error? How is it
best done using trial and error?

Thanks
Scott