# How to calculate rotation for three square interlocking angled legs?

I was inspired to try to create this style of legs for a small side-table that I am building:

However I have been running into some trouble trying to work out how to construct it.

I've been attempting to model it in SketchUp so that I can get all the measurements / angles worked out before I get to work, but I can't seem to figure out how to get the rotation of the legs right, so that the sides of each of the wooden strips on all three legs all lay flat against each other at the joints.

The first thing I attempted was to simplify things a bit and just model 3 straight square legs, intersecting right in the middle of each other... each leg is angled at 45° inwards. I had assumed that a matching rotation of 45° of each leg about it's lengthwise axis would get everything lined up, but it's ever so slightly off:

So the question is, how do I calculate the proper angle for the rotation of the legs, ideally for any given inward angle (ie: the legs will intersect closer to the top than the bottom, so are angled inwards at only 28°)?

• For most things like this the longitudinal axis of each of the three lengths is at 90° to the others. Don't know what's up with the model but if those legs are rotated correctly the overlapping area should be perfect, so it's likely a simple error or something wrong in SketchUp. Anyway, if you alter the intersecting angle it hugely complicates things since you're then into compound-mitre cuts which most find so difficult they avoid doing them a second time after they've done them once LOL Commented Jan 6, 2018 at 8:07
• If it is just SketchUp that's to blame that would be both a relief and a big annoyance! (I did spend a few hours reconstructing the model multiple times to make sure it wasn't user error, as that was my first assumption haha). As for the more complicated angle... I do tend to make things very difficult for myself - but the result is usually more interesting! Sometimes it's worth the struggle, I'm hoping that will be the case here Commented Jan 6, 2018 at 23:58
• Well if someone doesn't come along with an Answer that helps here on how to do this at a shallower angle than a three-way 90° intersection (unfortunately highly likely now given the doldrums) I think your best bet would be to Google to see if you can find out how someone else has already done it. Always best to not reinvent the wheel if possible. I was going to suggest starting from how the joint is done at 90 and extrapolating from there but I suspect that won't work and instead a purpose-suited three-way joint would have to be executed. Commented Jan 7, 2018 at 9:57
• I can duplicate the problem, but I cannot figure out what I (or sketchup) is doing wrong. I think it is resolving the intersection incorrectly.
– Ashlar
Commented Jan 8, 2018 at 2:56
• Im going to add to this even though it was so long ago. This page has a great calculator rechneronline.de/pi/corner.php If you lookup cuboid it will help. In terms of the bevel, it is 35.24 degrees and a 45 degree mitre. Commented Jun 10, 2020 at 16:13

## 3 Answers

The legs must be perpendicular to each other avoid any 'diamonding'. See:

It is only at 90 degrees the intersection will maintain squareness. You can form your caltrop this way:

I think to to achieve squareness to each other and shift or 'choke' the intersection closer or further from the middle point you must 'taper' the overall geometry. I believe that is why the circumference of the bottom is smaller than the circumference of the glass in your example. It also may explain the broken lines to help disguise this fact.

This is a similar table base that I built a few years ago. The joint you trying to replicate is actually a Japanese joint that is most common in much smaller stock and used to make puzzles called “burr” puzzles... Japanese burr puzzle is where you’ll need to start. Burr puzzles are typically built using square stock, and in multiples of 3 creating this illusion that the 3 pieces of wood are going through each other, because they meet at 90 degrees but you cannot see the joint. some like to think it’s not actually 3 pieces of wood and there is some sort of internal hardware trickery taking place. This table I built is the most basic burr puzzle joint is a 3 piece burr, and from there they move up to 6,9,12.... so forth. You are right that the miter on the legs is a 45, and I believe you’re really close on the bevel at 28. If I can remember the compound miter I used was something like 45 miter and 25.5 bevel...that’s on both ends parallel, so the legs are flat against the glass and the floor. Good luck to you what youre trying to build was originally built to be mentally strenuous and challenging to put together alone... building it is a whole other monster. zooming in on your picture it looks like you have possibly a 12 piece burr puzzle joint you’re trying to build. You are a brave man. hope this helps!!! Your best bet is to just get a bunch of square stock and start practicing your notches. Once you have all the pieces notched, the fun part is putting the puzzle together!!!

I came across a similar design recently.

The pictures on that site show (as @Graphus mentioned) that the 3 "legs" are all at right angles to each other.