When my son asked me to make him a couple of gymnastic rings so he could add then to his exercise routine, I said "Sure". At first I was thinking super simple, but after thinking about it for a while, it's not as easy as I thought. The first question was what material would work the best for these gymnastic rings, solid wood, laminate some wood together? I thought this would be a perfect project for Baltic Birch plywood. It's solid, won't move much, and it will be really strong. Strong was the key especially if you are going to be hanging from them. Won't be good at all if they broke while doing some type of full body weight exercise. The next challenge wasn't the outer dimension because right away I knew the tables would be a good choice. It was the inner dimension that was kinda tricky. I thought I could just do it on the lathe, but I tried a test ring and I was taking me forever and quickly dulling my tools. I didn't care for that approach. Next, I thought drilling a hole and using a jig saw but not liking a jig saw, and knowing it wouldn't be accurate or straight up and down, that idea was out. Finally it came to me - a router with a circle jig would be perfect. Knowing I could reference off the same center point for both the inner and outer circle they would be concentric to each other and it would be safest way to cut the inner.
After getting them cut out, it was just a matter of rounding the edges with a 5/8" round over bit on the router table and a bunch of sanding. Even though it was Basic Birch plywood I still had a couple of voids which were easy to fill with some Starbond Ca glue-https://www.starbond.com/ , which worked out great. As for a finish on the ring, I did some research and most said no finish at all. One said that the oils on your hand will eventually create a finish.
So give these a try... It's a fun project and some how very satisfying to make a couple of rings out of wood. Be Safe.

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Straps for rings - https://amzn.to/3s10U0M
Titebond Glue - https://amzn.to/2OKWfBG
Porter Cable Router - https://amzn.to/3aqgsVA
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5/8" round over Whiteside bit - https://amzn.to/3u1SqIq
CMT Straight 3/8" bit - https://amzn.to/2ZlCtyP
Freud Table saw Blade great for ply wood - https://amzn.to/3jTwCdg