Any geometry geniuses up in here? Help solve my riddle please?
4I think I might have been napping when I covered geometry in high school. I basically have zero skillz. Any maths genius, please help!
Goal: Make a tube out of wood.
The tube needs to have a 21.5" inside diameter, 24" ouside diameter, and 1.5" height.
I will do this by making a polygon from bits of 2x4 lumber (1.5" x 3.5" actual) then using a router, round the inside and outside radius into the circular tube.
Challenge: Find the absolute minimum number of sides required to meet my dimensional requirement. Pent? Hex? Hep? Also, I need to know how long to cut my 2x4 bits, and at what angle.
I think this is probably easy for someone who remembers the formulas from high school geometry. That someone surely isn’t me!
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some questions.
a) vertical (top to bottom of the 1.5 inch dimension)
b) ‘horizontal’ (across the 24" dimension in straight lines)
c) ‘horizontal’ (across the 24"diameter but in segments around the edge)
3)what are you planning to attach the wood pieces together with… wood glue, nails, screws?
@chienfou
Clarification on #2: I plan on cutting the 2x4 bits along their length at whatever length and angle necessary and joining them end to end. I think I might get there with a heptagon or maybe a hexagon?
@chienfou not sure if that is totally clear, so here is a quick sketch (not to scale) to clear it up a bit maybe…
@chienfou Yup, option C, boards laying on their 3.5” width, making the polygon 1.5” high.
Looks like you’ll need 24 planks if you’re very careful.

@walarney That many? What length and angle?
@ruouttaurmind @walarney definitely less than that, but still working on the total number/angles.
@ruouttaurmind when you said tube, i was thinking of something longer/deeper than wide, so you’d be joining 2x4s side by side with end grain up. but if you’re going for more of a flat ring, laying the 2x4s on their wide side and joining end to end, I think you can do it with 5. But gluing end to end is a problem. Now if you stack a bunch of the rings to make a tube, you can stagger the joints…

angle is the easy part, it’s just 360/number of sides. length is trickier. I’m lazy and would use CAD, but I don’t currently have any handy.
Also, I don’t think you’ll have much pocket screw left after routing out the tube.
@walarney Flat ring. Definitely a more fitting word than “tube”.
Missed the 1.5" high in the original post. Depending on how strong you want this to be, you might still consider making two rings 3/4" tall and gluing them stacked with the joints overlapped. Regardless of which layout you use.
No clue if this is right or helps

I came up with a formula for the number of sides. I get 180 / cos-1((outer radius - board width) / inner radius). Solving with your inputs, I get ~4.7 sides, so a pentagon. The outside side length formula I came up with is (outer diameter * tan (36 deg)), which evaluates to very nearly 17 7/16". The angles will be 54 deg.
@mschuette So if I follow what you have, if he makes each piece 17 1/2" to 17 3/4" on the long side, with a 36 degree cut off off square,he should still be able to get the inner and outer dimensions out of those pentagons with a bit of play to spare, correct?
@chienfou Yep, 36° off square. Imagine dividing the pentagon into triangles. Each inner angle is 72°. Now drop a perpendicular to a side. That right triangle has a 36°, 90°, and then 54° angle. That 54° is your cut. It’s likely to be 36°, though, depending on the markings on the saw. The outsides should be a hair shy of 17 7/16" and any problems with the angles or lengths will really multiply quickly.
Given all that, built with the 2x4s laying down, he can make a ring up to 1.49" (he needs only 1.25") thick by removing the outer points and rounding the inner sides. From a practical standpoint, two 3/4" tall rings glued together with joints interleaved might be worth the effort since I’m not sure the pocket screws can be positioned wholly inside the ring portion.
*Edit: I got the trig right, I just messed up the subtraction!
@chienfou And I did my angle math wrong… 54° + 36° = 90°, of course! I will update that, just wanted to call myself out.
You could also do a heptagon with 1x3 or a 15-gon with 1x2 (or 2-by, but @walarney was absolutely right about the two rings stacked). I would avoid heptagons at all costs, as those angles are messy. A 15-gon actually works out pretty nicely, uses almost a foot less lumber and requires removal of less material, but it sure is a lot more gluing! It also sounds like you have some 2x4 ends available (don’t we all?!)
Going the other way, 1x6s let you make a square which will have a ton of advantages for building it, but requires exactly 8 linear feet, and with saw kerfs, you aren’t going to get that out of an 8’ board.
@chienfou @mschuette @walarney
This is basically what I’m after. The yellow is representative of the “flat ring” I want to mill out of the polygon. ID of the ring should be 21.5", OD should be 24", leaving me a ring/tube wall thickness of 2.5" and a height of 1.5":
As far as the pocket screws, I was thinking if I align them on the centerline of the piece I’m joining to, that might give the clearance I need to wind up fully within the ring/tube. Though even if I only use the pocket screws to secure the mess until the glue is fully cured, then back them out, I should still be golden.
In this image, the red cone represents the alignment of the screw. In this case I scaled a 2" screw, but could probably safely go down to a 1.5" screw and still get a good bite on the target piece. My pocket bit likes about 1/2-inch of material to remain at the bottom of the drilled hole, so a 1.5" screw would give 1" of penetration into the target piece. Between glue and a pocket screw I’m confident in the strength of the joint. Better still if I do a 2-layer ring as some of you recommended.
@chienfou @ruouttaurmind @walarney If the OD is 24" and the wall thickness you need is 2.5", then your ID is 19" (two wall thicknesses across the OD). I think the pocket screws CAN work, but if you hit one with your router, that’s on you!
@mschuette @ruouttaurmind @walarney
I bet that changes the number of pieces he will need from 5 to 6 at least.
@chienfou @ruouttaurmind @walarney It’s 7, actually. The dreaded sevenths. You could up it to an octagon for sanity. As a bonus, most mitre saws have a detent at 22.5°
@ruouttaurmind so… which is it, 1.25 inch or 2.5 inch???
and yeah… I would go with the octagon
@chienfou I’m proving my geometry ignorance again. The ring wall thickness… not the height, should be 2.5”. The height will be 1.5” as dictated by the thickness of a 2x4.
So here’s what I’m not grasping: if the ID is 21.5”, and I add 2.5” of thickness, I get 24” OD. Where am I screwing up my maths?
@chienfou NVM… I just caught my gaff. 2x the thickness. Duh. Because there is 2.5” on EACH side of the circle.
@mschuette
Yup, mine has detents at 45, 33 and 22.5.
@ruouttaurmind don’t guess you need any more suggestions, but maybe put the screws in from the bottom instead of the sides. Easier to avoid router bit damage.
I don’t think this is a tube as much as it is a ring.
Also, I was told there would be no math.
@therealjrn
glad you agree with the comments above
Okay, @chienfou.
Watch how you post from now on, don’t be repeating anybody and make sure you read every single thread and every single post.
What’s your plan for holding the router/wood steady while you make the inside/outside surface curves? You won’t have a super long edge glued surface to resist any push you might make with the router… another good reason for stacked rings especially with grain shifts. (think plywood)
@chienfou a 2’ square of plywood and a jigsaw would be FAR easier!
@mschuette certainly for the initial shaping then cleanup with a router…
@chienfou @mschuette My issue with plywood is the inevitable voids. I could get some high end Baltic birch and almost certainly escape the dreaded voids, but that stuff is a bit above my pay grade for this project.
I intend to make a circle jig for the router with a variable pivot hole. I’ll use a 2” trim bit to shape the circle. Bolt the jig to the router plate, secure it in the center of the polygon and swing away. To keep the polygon from walking I will pound some dogs and holdfasts around the perimeters, moving each holdfast as I come to that spot with the router, then replacing it once I’ve trimmed past it. Slow and steady pace, taking maybe 1/16” on each pass, budging the jig length, around again, lather, rinse, repeat until I’ve reached my target dims.
@mschuette @ruouttaurmind sounds like your have it planned out well. Post pics when you are done please…
On a slightly different topic - what do you need this for?
@Kidsandliz agree! My bet is another project for his mom.
@tinamarie1974 I bet you are right.
@Kidsandliz In another topic I posted some pics of my concrete planter experiments. Everything looks great on the last few samples… except for the rim.
Solution? Build this wood ring, cut a channel into the ring and incorporate it into my mold jig to shape the rim into a smooth and consistent design. My simple 2 piece mold becomes a simple 3 piece mold with a nice contoured rim.
Or at least that’s my vision.
On this one I spent a few hours honing the shape of the rim with a brick. Too. Much. Work.
@ruouttaurmind Cool. Post one when you are done. But how are you going to finish it so that when people spill water on it then it won’t look water stained.
@Kidsandliz @ruouttaurmind could you use a dremel to smooth the edges, or some kind of electric sander? The brick sounds brutal
@Kidsandliz it’s an outdoor planter. It will weather and recover just like a concrete sidewalk.
@ruouttaurmind You making it out of cedar?
@tinamarie1974 Concrete is hard. Well, hard as concrete.
It will scoff at most abrasives. It would laugh at anything I could clamp into my little rotary tool. Also, it’s pretty huge. 24” across. That’s a lot of dremeling. 
@Kidsandliz The planter? It’s concrete. The wood rim is only going to be a mold to shape the concrete as it’s poured and cures. Then the mold is removed and all you have left is 100% concrete.
@ruouttaurmind well darn, I was hoping I had a genius idea that was going to save you time. Oh well…
Good luck with the wood mold
@tinamarie1974 I appreciate your input!
@ruouttaurmind @tinamarie1974
Link please? Google failed me!
@ruouttaurmind @tinamarie1974
oh… and have you tried using “hypertufa” ?
@chienfou
Here, here, and here.
Yup, I messed with it after @msklzannie posted a link about it in another topic. The result was not good. In fact it was so awful I was too ashamed to even post a pic. It’s still on my radar, but I will need some experimentation to get it to work as designed. This weekend I am going to try adding sawdust and cast a small version of my concrete bowl to see how that goes. I’ve read that it has a similar effect as hypertufa. Sawdust seems to breed in my shop, so I’ll have an endless supply if that works out ok.
@msklzannie @ruouttaurmind Thanks for the link(s). I had a friend at work that was making hypertufa planters so it’s been on my radar as a “gotta try that… someday” project.
@chienfou My first concrete planter was super easy and dirt cheap. I used a Hyper Tough 30 gallon trash can for the mold ($10 @ Walmart), a 60 pound bag of Quikrete High Strength concrete mix ($4 @ Home Depot) and some non-stick cooking spray for the mold release agent. Total prep time was about 30 minutes, including making and prepping the mold, mixing and pouring the concrete.
Since then I’ve learned that Crisco shortening makes a much better release agent. I can hit the mold with a heat gun for a few second, the Crisco liquefies, and the mold practically leaps off.
@ruouttaurmind what do you use for the inside (to keep the concrete in place)?
@ruouttaurmind D’oh… just looked closer at the pics… You cut it in half and stuck the “bottom” inside the top, didn’t you?
@chienfou Yup, that’s what I did.
I decided I wanted to make a planter 14” tall. I took a WAG that I would need the bottom thickness of the planter to be about 1.5”. Based on absolutely nothing more than “ya, that seems about right”.
I clamped a sharpie to the leg of my work bench at 12.5” off the shop floor. With the trash can upright, I held it firmly on the floor and rotated it around the sharpie to draw a reasonably even cut line.
Using a 4.5” grinder with a cutting wheel, I carefully cut along the line. Dremel tool, trim router, even a jigsaw would probably do fine. The bottom half of the trash can was ready to become the inner mold shell for the planter. The top half would be the outer mold shell.
I had a scrap 30”x30” piece of melamine MDF, but just about any old scrap of sheet material will do just fine. I turned the inner shell mold upside down onto the MDF and glued it in place with hot glue. Liquid Nails, silicone, bondo… just about anything with decent adhesive properties would do the job. I took care to totally seal it to the MDF so concrete wouldn’t leak inside the mold shell, and to contain the water which will leach from wet concrete.
Next, I turned the outer shell upside down and centered it around the inner shell. In my case I just eyeballed it. Close enough was good enough.
Once the outer shell was centered, I used short drywall screws and secured the outer shell to the MDF. I used screws rather than adhesive because you have to be able to easily remove the outer shell after the concrete sets up. I also ran a thin bead of silicone around the point where the rim was touching the MDF. Again, to contain any leaks.
Finally, I coated the inside of the mold with cooking spray. Bad plan. In future tests I used Crisco shortening, and lube up the mold shells before securing the outer shell in place. MUCH easier to get all the bits coated. To clarify, you want to coat the OUTSIDE of the inner shell and the INSIDE of the outer shell.
That’s it! You’re ready to mix and pour the concrete. In my last couple tests I switched to type S mortar and that seems to be working well. It’s a couple bucks more per bag, but it’s much easier to get down into the void between the mold shells without having to make your mix too soupy.
As I’m filling the mold with concrete, I use a sawzall without a blade, and press the sawzall foot against the MDF to vibrate everything and help settle any bubbles and voids. Add concrete, vibrate, more concrete, vibrate, etc.
I added concrete until the inner mold was about 1.5” under the top level of the concrete. Smoothed the surface with a piece of scrap 1x2, and washed up my tools.
I let it sit for about six or seven hours, then removed the outer mold. The amount of time you let it sit depends heavily on how soupy you mix your concrete. Mine was about the consistency of smashed potatoes. Well, maybe slightly thin smashed potatoes. Too thick and you won’t be able to work it down between the mold walls. Too thin and it affects the strength of the concrete, and takes forever to cure.
After removing the outer mold I let the whole thing sit overnight before removing the inner shell. And there it was, my first concrete planter.
@ruouttaurmind thanks for the tutorial. I guess you don’t leave any drain holes in the bottom, correct.
@chienfou Drain holes in the planter? Or drain holes in your mold?
Adding a drain hole in the planter is easy peasy. Place a length of pipe or tubing on top of the inner mold. Secure it in place with glue, caulking, silicone or whatever so when you pour the concrete your drain pipe stays in place. After the concrete firms up (but before it cures - a few hours should be good) you can remove the pipe, or just leave it in place and trim it flush with the bottom of the planter. Add as many as you feel like you need.
Adding drain holes in the molds so excess water can drain? Bad plan. Concrete will work into the holes and grab your mold making it difficult to impossible to remove without destroying the mold.
@ruouttaurmind nope, I was talking about in the planter. I figured you could even just push in pieces of dowel, broom handle etc. and if there was any concrete under that it would be so thin it would be easy to break out after it was set. Again, pulled once the concrete set but before they got ‘locked in’
@chienfou So did you become curious enough to add this to your weekend activities list?
@ruouttaurmind Yes, absolutely. But not THIS weekend.
Taking the grands camping in about 10 days so trying the get gear that hasn’t been used in several years clean, checked out and ready.
Set up a couple of dome tents to air out and was going to spray them down with disinfectant to kill the mildew smell but due to people freaking out about COVID19 there was none to be had locally in my small town.
Guess it worked out OK though. One of the tents blew into the pool overnight so at least that one is clean!
Do a search for Stave Drum builds.
@bxm83 Wow. Those make my little project look like a stroll through the park.
https://www.instructables.com/id/How-to-Make-a-Stave-Snare-Drum/
Might be a little more than you expected but this is the general idea you were going for right?
@bxm83 That’s the process.