Gazebo Designs

Help with geometry for a gazebo roof?

A hexagonal gazebo roof, in plan, shows six equilateral triangles. The slope of the roof is 20 degrees. So the actual angles of the pieces of plywood roofing are not 60 degrees. Along each ridge there is a metal flashing which receives the siding and keeps the joint from leaking. How do I figure out the angle between two adjacent roof panels, along this line? To visualize, it is the angle at which the metal flashing will be folded to cover the joint. I know the builders will just bend it till it fits. Still, I want to know! Thanks!

Public Comments

1. sharrrkrkkkkkies
2. Lets call the hexagon side x. A perpedicular from the center of the hexagon onto a side (call this perpendicular OM) is x*sqrt(3)/2 Call the meeting point of the roof R. There is a theorem that says : If RO is perp to the plane and OM is perp to x then RM is also perp to x. RM=x*sqrt(3)/2*cos(20) The sides of the roof will be: Sqrt(RM^2+x^2/4). To measure an angle we must draw 2 perps on the common line of 2 planes. Two adjacent triangles will have meeting heights. x*RM=h1*side h1=x*RM/side. You have a triangle formed by h1 by h2 (h2=h1) and by a line in the hexagon which is x*sqrt(3) h1=x*x*sqrt(3)/2*cos(20)/sqrt(3/4*x^2*cos(20)^2+x^2/4) h1=x*sqrt(3)*cos(20)/2 / sqrt(3/4cos(20)^2+0.25) sin (unknown angle/2) = x*sqrt(3)/2 / h1 = sqrt(3/4cos(20)^2+0.25)/cos(20) unknown/2 = 39.7 degrees. unknown = about 80 degrees