How to calculate width and depth to determine measurements using math (geometry?) along a cone shape [closed]

Question

all. I get accused of being too wordy with my posts but I always feel like I need to provide all the details. If you want to go right to my question, just look for "So, my question is..." in bold toward the bottom to get right to it.

Quite embarrassed that I need a refresher on what is probably high school level geometry. I'm in my fifties and was never good at math to begin with. It all went in one ear and out the other. To get to it I'm constructing two identical 10' tall cone-shaped Christmas trees. It's for a vintage modern home and will be pretty much simple, and literally conical in shape. However, it's not a perfect circle. It's elliptical (to save living room floor space). I decided to use rebar because it's cheap and is the ideal length for my very tall trees. My idea is to create wooden elliptical rings and use four rebar rods per cone. I've cut out a notch where I am placing the rebar and holding it in place with galvanized wire. You can see it all in my photo I've provided here.

The bottom ring is 36" wide with a depth of 21.5. I want to place another four wooden elliptical rings, which I'll cut and wire at 2ft intervals going upwards.

Here's why I want to do math to determine the size of my other elliptical rings. At this point, the rebar rods are so flexible that they're sagging toward the middle so I can't get the outer measures to create my other elliptical rings. I also do have some type of straight edge that's long enough to pull the rights straight in order to take the measurements. Not to mention, it would probably take three people to hold the rights perfectly straight while the third person takes the measurement for both major and minor axes.

So, my question is what geometry calculation do I use to determine the sizes of my other rings? If the top (where I'll place the star) is considered to be zero and since my existing bottom elliptical's major axis is 36" and the minor axis is 21.5", and if I move up in 2ft increments, how do I figure out the sizes for my ellipses at 8ft, 6ft, 4ft, and 2ft from the top?

I'm not expecting someone to do all the math for me, but can someone point me in the right direction on how to create either the next ellipse at 8ft (2ft from the bottom) or the 2ft ellipse (measuring from the top downward)? A theorem, perhaps?

Image posted here to provide a visual of what I've created so far:

Answer 1

You’re very much on the right track with the 0 at the top and the 36”/21.5” at the bottom. The sides of the tree will be straight lines, so you can interpolate linearly to find the ellipse size at any height. For a 10’ tree, each 2’ in height will add 1/5 in width and 1/5 in depth.

So, your total widths/depths will be:

• 8’, 7.2”/4.3”
• 6’, 14.4”/8.6”
• 4’, 21.6”/12.9”
• 2’, 29.8”/17.2”.

To be more precise, don’t forget the width of the rods - they will add a constant to each layer that you must remove from your elliptical rings’ thickness.

The geometry theorem to prove this is the “angle-angle similarity theorem,” which states that two triangles with two identical angles will be similar. Corresponding sides of similar triangles always have the same ratio.

Answer 2

Can't answer the math easily. However, I have a suggestion:

Instead of working on the tree with the base on the floor, suspend it from the top - e.g., put a hook in the ceiling and hang it from that. Gravity should help keep the rods straight - if not, add a little weight to the bottom ring. Then measure at the desired height and you're done.