Thread: Triangles connecting a box to a cylinder

1. Triangles connecting a box to a cylinder

This is related to the last problem I posted, but should I believe be much simpler: geometry rather than multivariable calculus. It is nonetheless a bit tricky to wrap my head around.

Here is the object in question:

See the purple parts?

The idea here is that it's a connector (converter) linking a cylinder to a box. The cylinder's base is a circle with diameter 280mm, and the box is 200mm by 200mm. They are separated by a distance of 200mm. Their bottoms line up.

They are connected by four trapezoids (the base of each is a full side of the square end of the box; the top of each is very narrow and connects to the top of the cylinder like so:

The red is welding material. In other words, the trapezoid connects right onto the very top of the circle.

After that, a series of 36 triangles (9 projecting from each corner of the box) extend out to approximate the circle. Each hits the circle with a base of 21.98mm.

I did some geometry and found that the first triangle to the immediate left of the top trapezoid has a base of 21.98 and shares a side with the left side of the trapezoid of 233.08mm. The angle opposite the 21.98mm base is 9 degrees. It is not a right triangle, as far as I can tell, so the Pythagorean theorem and mighty chief SOH CAH TOA both seem powerless to determine the other side of the triangle. (My plan was to find that, then use a chain process to get the next triangle and the next triangle and so on.)

Any ideas for efficient ways to do this? The goal is to get the area of the triangular joiners...and then the volume of the joining unit. (Think of it as a boxy pipe connected to a cylindrical pipe.)

Thanks again. Man, I wish I had done more math in college...

EDIT: Here's a diagram with the desired segment highlighted in red. This would be a top-view.

2. Okay, I think I've made a little progress...I just realized it's a regular 40-gon overlying the circle.

(Except it differs from the above in that it's resting on a side, not on a point. Just a slight rotation.)

This at least gives the question some more structure...what I'm trying to do, I guess, is figure out how much each point differs from the other in horizontal and vertical coordinates. If I can get that, I can do an x, y, z to x2, y2, z2 distance calculation and get the sides of the triangle.

Any idea what the rate of change is? Like, how much does it go up for each subsequent side of the 40-gon?

3. Okay, nevermind. I solved it, I think...the trick was realizing that the internal angles of a regular polygon are calculable and equal.

I feel so accomplished!