Angles in a Hexagonal Pyramid
Date: 07/25/2002 at 09:34:14 From: Chris Subject: Angles of a hexagonal cone Dr. Math: The pitch of the roof of a gazebo is 45 degrees, and the entire roof can described as a hexagonal pyramid. My question is this: What is the interior angle that the planes of the pyramid make with each other? Or, what is the interior angle of the intersection of two roof panels and how does this angle change with the pitch of the roof? Obviously, if the pitch of the roof were 0 degrees, the angle of intersection would be 180 degrees. And if the pitch of the roof were 90 degrees (vertical roof), the angle of intersection of the roof panels would be equal to 60 degrees (the interior angle of a 2D hexagon). For a roof pitch between 0 and 90 degrees, can you assume a linear relation for the angles of intersection? Also, how do these angles change along the length of the roof (or height of the pyramid)? I hope that this is not too involved and I really appreciate your time. Thanks, Chris
Date: 08/01/2002 at 02:16:58 From: Doctor Douglas Subject: Re: Angles of a hexagonal cone Hi, Chris, If you have a pyramid whose base is a regular hexagon (all sides and all angles equal), and the apex is located directly above the hexagon center, then the dihedral angle (the angles that the faces make with each other) is given below. If the hexagon has sides of length A and has height H, then the ratio z = H/A can be thought of as a sort of "slope" (although it is actually the slope of the edge from base vertex to the point). The angle that you are seeking is [sqrt(3)/2] sqrt(1+z^2) z x = arcsin ------------------------- z^2 + 3/4 where 90 deg <= x <= 180 deg This is not a linear relation, but you can check that it has the proper "limiting" behavior. If z is very large (i.e., you have a very pointy pyramid), then the sides are like those of a hexagonal prism, and x becomes arcsin(sqrt(3)/2). A calculator yields the result of 60 degrees here, but we have to take the supplementary angle, or 180 - 60 = 120 degrees. This answer agrees with our reasoning from the hexagonal prism, which has faces that meet at 120 degrees. If z is very small, or approaches zero, then x approaches 180 degrees (recall that x is in the range 90 deg to 180 deg), which again agrees with our intuition of a squat pyramid whose faces are nearly parallel. The computation of the formula for x above is found from good old three-dimensional geometry, and the use of the three-dimensional vector cross product. I'll sketch out where it comes from, and you can work through it if you're interested in the details. 1. z = H/A is the only parameter we need. The angles are independent of the actual size of the pyramid, so we might as well choose a pyramid with side A = 1, and height H = z. 2. Let the base of the pyramid be centered on the (x,y) plane, and let one of the vertices be at (x=0,y=1). We need to compute the direction of two normals (directions perpendicular to the faces). We can compute these normals at any points on two adjacent faces. 3. Let the first such point be at (x=sqrt(3)/2, y=0, z=0), at the midpoint of one side. The normal to the plane here is in the direction (z, 0, sqrt(3)/2). The length of this normal is sqrt(z^2 + 3/4). 4. The normal to the adjacent plane will be essentially the same, but rotated in the xy plane by +60 degrees counterclockwise. This direction is (z/2, z*sqrt(3)/2, sqrt(3)/2). 5. The vector cross product between these two directions is (-3z/4, sqrt(3)*z/4, z^2*sqrt(3)/2). The length of the vector cross product is the square root of the sum of the squares of the 3 components, or sqrt(9z^2 + 3z^2 + 12z^4)/4 = sqrt(12)sqrt(z^2 + z^4)/4 = 2*sqrt(3)*z*sqrt(1 + z^2)/4 = sqrt(3)*z*sqrt(1+z^2)/2 6. The vector cross product length is also equal to the product of the lengths of the two original vectors times the sine of the angle that we seek: sin(x)*sqrt(z^2 + 3/4)*sqrt(z^2 + 3/4) = sin(x)*(z^2 + 3/4) 7. Equating these last two expressions allows us to solve for x to get the equation that I gave above. We need to take account of the fact that the cross product gives us the smallest angle between the two normals, and in your application, we're typically looking for the supplementary angle to this. I hope this helps answer your question! - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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